# Time dispersion in quantum mechanics

John Ashmead  Visiting Scholar, University of Pennsylvania, USA
<EMAIL_ADDRESS>
###### Abstract

In quantum mechanics the time dimension is treated as a parameter, while the
three space dimensions are treated as observables. This assumption is both
untested and inconsistent with relativity. From dimensional analysis, we
expect quantum effects along the time axis to be of order an attosecond. Such
effects are not ruled out by current experiments. But they are large enough to
be detected with current technology, if sufficiently specific predictions can
be made. To supply such we use path integrals. The only change required is to
generalize the usual three dimensional paths to four. We predict a large
variety of testable effects. The principal effects are additional dispersion
in time and full equivalence of the time/energy uncertainty principle to the
space/momentum one. Additional effects include interference, diffraction, and
entanglement in time. The usual ultraviolet divergences do not appear: they
are suppressed by a combination of dispersion in time and entanglement in
time. The approach here has no free parameters; it is therefore falsifiable.
As it treats time and space with complete symmetry and does not suffer from
the ultraviolet divergences, it may provide a useful starting point for
attacks on quantum gravity.

> “Wheeler’s often unconventional vision of nature was grounded in reality
> through the principle of radical conservatism, which he acquired from Niels
> Bohr: Be conservative by sticking to well-established physical principles,
> but probe them by exposing their most radical conclusions.” – Kip Thorne
> [1].
>
> “You can have as much junk in the guess as you like, provided that the
> consequences can be compared with experiment.” – Richard P. Feynman [2]

##  1 Introduction

###  1.1 Should the wave function extend in time as it does in space?

In relativity, time and space enter on a basis of formal equivalence. In
special relativity, the time and space coordinates rotate into each other
under Lorentz transformations. In general relativity, the time and the radial
coordinate change places at the Schwarzschild radius (for instance in Adler
[3]). In wormholes and other exotic solutions to general relativity, time can
even curve back on itself as in Gödel or Thorne [4, 5].

But in quantum mechanics “time is a parameter not an operator” (Hilgevoord [6,
7]). This is clear in the Schrödinger equation:

$\imath\frac{d}{d\tau}\psi_{\tau}\left(\vec{x}\right)=\hat{H}\psi_{\tau}\left(\vec{x}\right)$
(1)

Here the wave function is indexed by time: if we know the wave function at
time $\tau$ we can use this equation to compute the wave function at time
$\tau+\epsilon$. The wave function has in general non-zero dispersion in
space, but is always treated as having zero dispersion in time. This would
appear to be inconsistent with special relativity.

Figure 1: Wave functions for Alice and Bob

Consider Alice in her laboratory, her co-worker Bob jetting around like a
fusion powered mosquito. Both are studying the same system but with respective
wave functions:

$\begin{array}[]{c}{\psi^{\left({Alice}\right)}}\left({{t_{Alice}},{x_{Alice}}}\right)\hfill\\\
{\psi^{\left({Bob}\right)}}\left({{t_{Bob}},{x_{Bob}}}\right)\hfill\end{array}$
(2)

Alice and Bob center their respective wave functions on the particle:

$\begin{array}[]{c}\left\langle{t_{Alice}}\right\rangle=\left\langle{t_{Bob}}\right\rangle=0\hfill\\\
\left\langle{x_{Alice}}\right\rangle=\left\langle{x_{Bob}}\right\rangle=0\hfill\end{array}$
(3)

These are distinct wave functions but give the same predictions for all
observables. And do so to an extremely high degree of reliability.

But at Alice’s time zero, Bob’s wave function extends into her past and
future. And at Bob’s time zero her wave function extends into his past and
future.

There are at least two problems here.

One is that in quantum mechanics there is a strict “plane of the present”. The
quantum mechanical wave function is non-localized in space but is strictly
localized in time. What if Alice decided to work with Bob’s wave function,
rather than her own? She will get by hypothesis all the same predictions, but
will be using a wave function that from her point of view slops into past and
future.

The other is that from the point of view of special relativity, there should
not be a strict “plane of the present” in the first place. We should be able
to rotate between the four dimensional references frames of Alice and Bob as
easily as we rotate between references frames for the three space dimensions.

What happens if we shift to four dimensional wave functions?

$\psi\left(\vec{x}\right)\to\psi\left(t,\vec{x}\right)$ (4)

Assume the coordinate systems for Alice and Bob are related by a Lorentz
transformation $\Lambda$:

${x^{\left({Bob}\right)}}=\Lambda{x^{\left({Alice}\right)}}$ (5)

Then their wave functions can be related by a Lorentz transformation of their
coordinates:

${\psi^{\left({Bob}\right)}}\left({x^{\left({Bob}\right)}}\right)={\psi^{\left({Alice}\right)}}\left({\Lambda{x^{\left({Alice}\right)}}}\right)$
(6)

and matters are much more straightforward.

We make this our basic hypothesis: the quantum mechanical wave function should
be extended in the time direction on the same basis as it is extended along
the three space dimensions.

We are playing a “game of if” here: we will push the idea as hard as we can
and see what breaks. We are not going to argue that this is or is not true. We
are going to look for experimental tests and then let the experimentalists
decide the question.

There are two principal effects:

  1. 1.

Dispersion in time appears on same basis as dispersion in space. Physical wave
functions are always a bit spread out in space; they will now also be a bit
spread out in time.

  2. 2.

The uncertainty principle for time/energy is treated on same basis as the
uncertainty principle for space/momentum. If a particle’s position in time is
well-defined, its energy will highly uncertain and vice versa.

####  1.1.1 Dispersion in time

If the wave functions normally have an extension in time then every time-
specific measurement should show additional dispersion in time.

Suppose we are measuring the time-of-arrival of a particle at a detector.
Define the average time-of-arrival as:

$\left\langle{\tau^{\left({TOA}\right)}}\right\rangle\equiv\int\limits_{-\infty}^{\infty}{d\tau}\tau{p^{\left({TOA}\right)}}\left(\tau\right)$
(7)

with an associated uncertainty:

$\left\langle{\Delta{\tau^{\left({TOA}\right)}}}\right\rangle\equiv\sqrt{\int\limits_{-\infty}^{\infty}{d\tau}{\tau^{2}}{p^{\left({TOA}\right)}}\left(\tau\right)-{{\left\langle{\tau^{\left({TOA}\right)}}\right\rangle}^{2}}}$
(8)

The probability distribution for the particle will normally be spread out in
space, so its arrival times will also be spread out, depending on the velocity
of the particle and its dispersion in space.

But if it also has a dispersion in time, then part of the wave function will
reach the detector – thanks to the fuzziness in time – a bit sooner and also a
bit later than otherwise expected. There will be an additional dispersion in
the time-of-arrival due to the dispersion in time.

As we will see (subsection 4.3), at non-relativistic speeds, the dispersion in
the time-of-arrival is dominated by the dispersion in space, so this effect
may be hard to pick out. At relativistic speeds, the contributions of the
space and time dispersions can be comparable.

####  1.1.2 Uncertainty principle for time and energy

Of particular significance for this work are differences in the treatment of
the uncertainty principle for time/energy as opposed to that for
space/momentum.

In the early days of quantum mechanics, these were treated on same basis. See
for instance the discussions between Bohr and Einstein of the famous clock-in-
a-box experiment [8] or the comments of Heisenberg in [9].

In later work this symmetry was lost. As Busch [10] puts it “… different types
of time energy uncertainty can indeed be deduced in specific contexts, but …
there is no unique universal relation that could stand on equal footing with
the position-momentum uncertainty relation.” See also Pauli, Dirac, and Muga
[11, 12, 13, 14].

There does not appear to be any experimental test of this or observational
evidence for it; it is merely the way the field has developed.

That is not to say that there are not uncertainties with respect to time, but
they are side effects of other uncertainties in quantum mechanics. For
instance, if a particle is spread out in space, moving to the right, and going
towards a detector at a fixed position, its time-of-arrival will have a
dispersion in time. But this is a side-effect of the dispersion in space.

Now consider a particle going through a narrow slit in time, for instance a
camera shutter. Its wave function will be clipped in time. If the wave
function is not extended in time, then the wave function will merely be
clipped: the resulting dispersion in time at detector will be reduced.

But if the wave function is extended in time and the Heisenberg uncertainty
principle applies in time/energy on the same basis as with space/momentum,
then an extremely fast camera shutter will give a small uncertainty in time at
the gate:

$\Delta t\to 0$ (9)

causing the uncertainty in energy to become arbitrarily great:

$\Delta E\geq\frac{1}{{\Delta t}}\Rightarrow\Delta E\to\infty$ (10)

which will in turn cause the wave function to be diffracted, to fan out in
time, and the dispersion in time-of-arrival to become arbitrarily great.

####  1.1.3 A necessary hypothesis

This question does not appear to have been attacked directly. As noted, the
assumption that the wave function is not extended in time seems to have crept
into the literature of its own, without experimental test or observational
evidence.

To make an experimental test of this question we have to develop predictions
for both branches:

  1. 1.

Assume the wave function is not extended in time. Make predictions about time-
of-arrival and the like.

  2. 2.

Assume the wave function is extended in time. Make equivalent predictions.

  3. 3.

Compare.

We have to develop both branches in a way that makes the comparison
straightforward.

Further, to make the results falsifiable we have to develop the extended-in-
time branch in a way that is clearly correct. A null result should show that
the wave function is not extended in time.

These objectives drive what follows.

####  1.1.4 Literature

The literature for special relativity and for quantum mechanics is vast. Our
focus is on the critical intersection of the two. References of particular
interest here include:

  * •

Stueckelberg & Feynman’s original papers: [15, 16, 17, 18, 19, 20].

  * •

Reviews of the role of time, Schulman, Zeh, Muga, Callender: [21, 22, 13, 14,
23].

  * •

Block universe picture: Parmenides, Barbour, Price: [24, 25, 26].

  * •

Cramer’s transactional interpretation: [27, 28, 29, 30].

  * •

The time symmetric quantum mechanics of Aharonov and Reznik: [31, 32].

  * •

The relativistic dynamics program of Horwitz, Piron, Land, Collins, and
others: [33, 34, 35, 36, 37, 38, 39, 40]. This program is a natural outgrowth
of Stueckelberg & Feynman’s work.

Our approach here may be understood as falling within the relativistic
dynamics framework, but with the emphasis placed on the “coordinate time”
rather than the “evolution parameter” aspects of that program. We will discuss
this in detail once an appropriate foundation has been laid.

###  1.2 Order of magnitude estimate

Has this hypothesis has already been falsified? Quantum mechanics has been
tested with extraordinary precision. Should associated effects have been seen
already, even if not looked for?

Consider the atomic scale given by the Bohr radius $5.3\ 10^{-11}m$. We take
this as an estimate of the uncertainty in space.

We assume the maximum symmetry possible between time and space. We therefore
infer that the uncertainty in time should be of order the uncertainty in space
(in units where $c=1$).

Dividing the Bohr radius by the speed of light we get the Bohr radius in time
$a_{0}=.177\ 10^{-18}s$, or less than an attosecond. $.177as$ is therefore our
starting estimate of the uncertainty in time.

Therefore from strictly dimensional and symmetry arguments, the effects will
be small, of order attoseconds. This is sufficient to explain why such effects
have not been seen.

At the same time, the time scales we can look at experimentally are now
getting down to the attosecond range. A recent paper by Ossiander et al [41]
reports results at the sub-attosecond level.

Therefore if we can provide the experimentalists with a sufficiently well-
defined target, the hypothesis should be falsifiable in practice.

###  1.3 Plan of attack

> Look, I don’t care what your theory of time is. Just give me something I can
> prove wrong. – experimentalist at the 2009 Feynman Festival in Olomouc

####  1.3.1 Primary objective is falsifiability

It is not enough to extend quantum mechanics to include time. It is necessary
to do so in a way that can be proved wrong. The approach has to be so strongly
and clearly constrained that if it is proved wrong, the whole project of
extending quantum mechanics to include time is falsified.

Our requirements are therefore that we have:

  1. 1.

the most complete possible equivalence in the treatment of time and space –
manifest covariance at every point at a minimum,

  2. 2.

consistency with existing experimental and observation results,

  3. 3.

and consistency between the single particle and multiple particle domains.

These requirements leave us with no free parameters. And having no free
parameters means in turn that our hypothesis is falsifiable in principle.

To get to falsifiable in practice, we will look for the simplest cases that
make a direct comparison possible.

We will also look at points of principle that need to be addressed, to ensure
that the approach is not ruled out by, say, violations of unitarity.

We will use the acronym SQM for standard quantum mechanics. We will use the
acronym TQM for temporal quantum mechanics. By TQM we mean SQM with time
treated on the system basis as space: time just as much an observable as the
three space dimensions.

We do not mean by “temporal quantum mechanics” that time itself comes in small
chunks or quanta! For instance, there has been speculation that time is
granular at the scale of the Planck time:
${t_{Planck}}\equiv\sqrt{\frac{{\hbar G}}{{c^{5}}}}\approx
5.39116x{10^{-44}}s=5.39116x{10^{-26}}as$. Maybe it is, maybe it isn’t. But as
this is 25 orders of magnitude smaller than the times we are considering here,
it is reasonable for us to take time as continuous. And since space is treated
by SQM as continuous, and since the defining assumption of TQM is the maximum
symmetry between time and space, we are required to take time in TQM
continuous.

####  1.3.2 Extruding quantum mechanics in time

If you have worked with a CAD/CAM or 3D drawing program you are familiar with
the “extrude” operation, where you take a circle or square and extrude it into
the third dimension111See Abbott’s classic Flatland [42] for a delightful
example.. That is all we are going to do here. We are going to use manifest
covariance as the rule for going from three dimensions to four – so that
${s^{2}}={t^{2}}-{x^{2}}-{y^{2}}-{z^{2}}$ is the invariant rather than
${r^{2}}={x^{2}}+{y^{2}}+{z^{2}}$ – but the principle is the same.

The advantage is that with this approach we have no free parameters. The
moment we introduce a free parameter we put falsifiability at risk since we
are thereby creating a “fudge factor”. And any fudge factor might let us say
“well perhaps no effect was seen at scale $x$, but at smaller scale $y$, then
we shall see!”. We want our experimentalists to be able to say, “not seen,
therefore not there!”

It might be that we should have done the extrude at a slight angle, extruding
our circle into, say, an oblate or prolate spheroid rather than a perfect one.
But this does not seriously impact falsifiability – such an oblate or prolate
spheroid should still have the same overall scale of a Bohr radius in time, it
will still give our experimentalist a fixed target. (We discuss possible
meanings of “a slight angle” in subsection 3.5 and again in the discussion).

We will use path integrals as our defining formalism. With the path integral
approach we can extrude the paths from three to four dimensions in a
straightforward way, while leaving the rest of the machinery essentially
untouched. With other formalisms, it is less clear what “extrude” means. But
once we have the meaning of extrude worked out for path integrals, we can, as
it were, rotate our path integral formalism into other formalisms, seeing what
“extruding quantum mechanics in time” looks like as a Schrödinger equation or
in quantum field theory.

Our plan:

####  1.3.3 Single particle case

In the single particle case we will:

  1. 1.

Generalize path integrals to include time as an observable.

  2. 2.

Derive the corresponding Schrödinger equation as the short time limit of the
path integral.

  3. 3.

Develop the free solutions. We will estimate the initial wave function, let it
evolve in time, and detect it. We will compute the dispersions of time-of-
arrival measurements in SQM and in TQM. In general the differences are real
but small.

  4. 4.

Analyze the single and double slit experiments. The single slit in time
experiment provides the decisive test of temporal quantum mechanics. In SQM,
the narrower the slit, the less the dispersion in subsequent time-of-arrival
measurements. In TQM, the narrower the slit, the greater the dispersion in
subsequent time-of-arrival measurements. In principle, the difference may be
made arbitrarily great.

####  1.3.4 Multiple particle case

We will then extend TQM to include the multiple particle case, i.e. field
theory. We will show, using a toy model, that:

  1. 1.

We can extend the usual path integral approach to include time as an
observable. The basis functions in Fock space extend in a natural way from
three to four dimensions, the Lagrangian is unchanged, and the usual Feynman
diagram expansions appear.

  2. 2.

For each Feynman diagram in SQM we can compute the TQM equivalent. Therefore
any problem that can be solved using Feynman diagrams in SQM can be solved in
TQM.

  3. 3.

The usual ultraviolet divergences do not appear (the combination of dispersion
in time and entanglement in time contain them).

  4. 4.

And that there are a large number of additional experimental effects to be
seen, including:

    1. (a)

wave functions anti-symmetric in time,

    2. (b)

correlations, entanglement, and interference in time,

    3. (c)

and forces of anticipation and regret.

####  1.3.5 Overall conclusions

With this done, we will argue in the discussion:

  1. 1.

that TQM is not ruled out a priori.

  2. 2.

that TQM is falsifiable. And given experimental work like Ossiander’s,
probably with current technology.

  3. 3.

that TQM is a source of interesting experiments. Every foundational experiment
in SQM has an “in time” variant.

  4. 4.

that TQM is a potential starting point for attacks on the quantum gravity
problem, since TQM is manifestly covariant and untroubled by the ultraviolet
divergences.

  5. 5.

that as TQM is a straight-forward extrapolation of quantum mechanics and
special relativity, experiments that falsify TQM are likely to require
modification of our understanding of either quantum mechanics or special
relativity or both. Something will have to break.

####  1.3.6 Limits of the current investigation

We do no more here than is required to establish that TQM is well-defined,
self-consistent, and falsifiable.

It will be clear from the development that the basic idea could be extended in
a number of directions; we review some of these in the concluding discussion.

The conventions used in this work are spelled out in A.

##  2 Path integrals

###  2.1 Overview

To extend quantum mechanics to include time we will take as our starting point
Feynman’s path integral approach to quantum mechanics [43, 44, 45, 46, 47, 48,
49, 50, 51, 52].

With the path integral approach, the only change we will need to make is to
generalize the paths from varying in three dimensions to varying in four.

To make clear what this means, consider the case of Alice walking her dog, say
from her front door to Bob’s.

Alice will take the shortest (classical) path from door to door.

But her dog will dart from side to side, now investigating a mailbox to the
left, now checking out a lamppost to the right. In fact, as a quantum dog he
will investigate all such paths simultaneously. While he will start at the
same time and place as Alice, and finish at the same time and place as Alice,
in between he will travel simultaneously along all possible paths.

But – in SQM – only along paths in space. At each tick of Alice’s digital
watch, her dog will be found off to the left or right, jumping up or digging
down, further along the path to Bob’s, or holding back for an important
investigation.

But in TQM, the quantum dog can – and therefore will – advance into the future
and drop back into the past. So that tick by tick of Alice’s watch, her dog’s
paths will have to tracked in four dimensions rather than three.

This is harder to visualize, being out of our normal experience. So we develop
the analysis a bit formally, letting math take the place of an as yet
undeveloped intuition.

Path integrals, as the name suggests, are done by summing over all paths from
starting point to finish, weighting each path by the integral of the
Lagrangian (the action) along it:

${\psi_{T}}\left({x_{T}}\right)=\int{\mathcal{D}{x_{\tau}}\exp\left({\imath\int\limits_{0}^{T}{d\tau
L\left[{x,\dot{x}}\right]}}\right){\psi_{0}}\left({x_{0}}\right)}$ (11)

Piece by piece:

  1. 1.

${\psi_{0}}\left({x_{0}}\right)$ is the initial wave function. We will be
breaking these down into sums over Gaussian test functions using Morlet
wavelet analysis.

  2. 2.

$\tau$ is the clock time as given by Alice’s digital watch. We will break up
the paths into the bits from one clock tick to the next.

  3. 3.

$\mathcal{D}{x_{\tau}}$ represents the paths. Each path is defined by its
coordinates at each clock tick. In SQM, these are the values of
$x_{\tau},y_{\tau},z_{\tau}$ at each clock tick. In TQM these are values of
$t_{\tau},x_{\tau},y_{\tau},z_{\tau}$ at each clock tick.

  4. 4.

$L\left[{x,\dot{x}}\right]$ is a suitable Lagrangian. We will be using one
that works equally well for both SQM and TQM.

  5. 5.

${\imath\int\limits_{0}^{T}{d\tau L\left[{x,\dot{x}}\right]}}$ is the action,
the integral over the Lagrangian taken path by path.

  6. 6.

And ${\psi_{T}}\left({x_{T}}\right)$ is the final wave function, the amplitude
for the dog to arrive at Bob’s door step.

We will look at:

  1. 1.

What do we mean by $\tau$ the clock time?

  2. 2.

What do we mean by the coordinate time $t$ in $t,x,y,z$?

  3. 3.

How do we define the initial wave function in a way that does not potentially
bias the outcome?

  4. 4.

What Lagrangian shall we use?

  5. 5.

How do we get the sums to converge?

  6. 6.

Having gotten the sums to converge, how do we normalize them?

  7. 7.

And what do all the pieces look like when we put them back together?

###  2.2 Laboratory time

We consider the action, the integral of the Lagrangian over time:

$\int\limits_{0}^{T}{d\tau\mathcal{L}\left[{x,\dot{x}}\right]}$ (12)

In classical mechanics, we are free to take the parameter $\tau$ as any
monotonically increasing variable. We will get the same classical equations of
motion in any case.

A typical choice is to select $\tau$ as the proper time of the particle in
question. However this makes it difficult to extend the work to the multiple
particle case, where there are many particles and therefore many proper times
in play.

Here we choose to use the time as shown on a laboratory clock. We take the
term laboratory time from Busch [10]. We will use the terms clock time and
laboratory time interchangeably.

It is useful to visualize this clock as a metronome, breaking up the clock
time into a series of $N$ ticks each of length $\epsilon$. If $\tau=0$ at the
source, and $\tau=T$ at the detector, we have:

$\epsilon\equiv\frac{T}{N}$ (13)

We will take the limit as $N\to\infty$ as the final step in the calculation.

###  2.3 Coordinate time

We visualize a four dimensional coordinate system coordinates $t,x,y,z$. We
will refer to $t$ as coordinate time by analogy with the three coordinate
space dimensions: coordinate $x$, coordinate $y$, and coordinate $z$.

Paths are defined with reference to this coordinate system. If the time by
Alice’s watch is $\tau$, then each path $\pi$ will have a location at $\tau$
given by:

${\pi_{\tau}}\left({t,x,y,z}\right)$ (14)

It may help to think of the coordinates as laid out on a piece of four
dimensional graph paper. At a specific clock tick $n$, a specific path $\pi$
will be represented by a dot on a specific vertex on the four dimensional
graph paper. If we want to see the progress of the path with respect to clock
time, we can flip the series of pieces of graph paper like one of those old
time flip movies.

If our graph paper has $M$ grid lines in each direction, the number of
vertices on a page is $M^{4}$, and the number of paths total is $M^{4N}$. Each
different sequence of grid points counts as a distinct path.

The path integral measure ${\cal D}x$ is usually defined by assigning a weight
of one to each distinct path, and then taking the limit as the spacing goes to
zero.

Since coordinate time is on the same footing as the three space coordinates it
has a corresponding energy operator:

${p_{x}}\equiv-\imath\frac{\partial}{{\partial x}}\Rightarrow
E\equiv\imath\frac{\partial}{{\partial t}}$ (15)

We will refer to this as coordinate energy. It is not positive definite or
bounded from below. Since $p_{x}$ can be positive or negative, by our
controlling requirement of covariance $E$ can be positive or negative.

We discuss the relationship between clock time and coordinate time in F.

###  2.4 Initial wave function

We need a starting set of wave functions $\psi_{0}$ at clock time $\tau=0$. We
will need wave functions that extend in both coordinate time and space. The
usual choices would be $\delta$ functions or plane waves.

In coordinate time these might be:

$\delta(t-t_{0})$ (16) $e^{-\imath E\left(t-t_{0}\right)}$ (17)

or in space:

$\delta(x-x_{0})$ (18) $e^{\imath p\left(x-x_{0}\right)}$ (19)

But neither $\delta$ functions nor plane waves are physical. Their use creates
a risk of artifacts.

More physical would be Gaussian test functions, for instance in coordinate
time:

$\varphi\left(t\right)\equiv\sqrt[4]{\frac{1}{\pi\sigma_{t}^{2}}}e^{-\imath
E\left(t-t_{0}\right)-\frac{\left(t-t_{0}\right)^{2}}{2\sigma_{t}^{2}}}$ (20)

or in space:

$\varphi\left(x\right)\equiv\sqrt[4]{\frac{1}{\pi\sigma_{x}^{2}}}e^{\imath
p\left(x-x_{0}\right)-\frac{\left(x-x_{0}\right)^{2}}{2\sigma_{x}^{2}}}$ (21)

But while Gaussian test functions are physically reasonable they are not
completely general.

We can achieve both generality and physical reasonableness by using a basis of
Morlet wavelets [53, 54, 55, 56, 57, 58, 59, 60, 61, 62].

Figure 2: Mother Morlet wavelet

Morlet wavelets are derived by starting with a “mother” wavelet:

$\phi^{\left(mother\right)}\left(t\right)\equiv\left(e^{-\imath
t}-\frac{1}{\sqrt{e}}\right)e^{\left(-\frac{t^{2}}{2}\right)}$ (22)

and scaling and displacing it with the replacement $t\to\frac{t-l}{s}$:

$\phi_{sl}\left(t\right)=\frac{1}{\sqrt{\left|s\right|}}\left(e^{-\imath\left(\frac{t-l}{s}\right)}-\frac{1}{\sqrt{e}}\right)e^{-\frac{1}{2}\left(\frac{t-l}{s}\right)^{2}}$
(23)

Any normalizable function $f$ can be broken up into wavelet components
$\hat{f}$ using:

$\hat{f}_{sl}=\int\limits_{-\infty}^{\infty}dt\phi_{sl}^{*}\left(t\right)f\left(t\right)$
(24)

And then recovered using the inverse Morlet wavelet transform:

$f\left(t\right)=\frac{1}{C}\int\limits_{-\infty}^{\infty}\frac{dsdl}{s^{2}}\phi_{sl}\left(t\right)\hat{f}_{sl}$
(25)

The value of $C$ is worked out in [62].

Therefore we can write any physically reasonable wave function $f$ in time in
terms of Morlet wavelets.

We may include space by using products of Morlet wavelets:

$\begin{array}[]{rcl}{{\hat{f}}_{{s_{t}}{l_{t}}{s_{x}}{l_{x}}}}&=&\int{dtdx\phi_{{s_{t}}{l_{t}}}^{*}\left(t\right)}\phi_{{s_{x}}{l_{x}}}^{*}\left(x\right)f\left({t,x}\right)\hfill\\\
f\left({t,x}\right)&=&\frac{1}{{{C^{2}}}}\int{\frac{{d{s_{t}}d{l_{t}}}}{{s_{t}^{2}}}\frac{{d{s_{x}}d{l_{x}}}}{{s_{x}^{2}}}{\phi_{{s_{t}}{l_{t}}}}\left(t\right)}{\phi_{{s_{x}}{l_{x}}}}\left(x\right){{\hat{f}}_{{s_{t}}{l_{t}}{s_{x}}{l_{x}}}\hfill}\\\
\end{array}$ (26)

Clearly it would be cumbersome to track four dimensional Morlet wavelets at
every step.

Fortunately we do not need to perform the Morlet wavelet analyses; we merely
need the ability to do so. As each Morlet wavelet may be written as a sum of a
pair of Gaussians, Morlet wavelet analysis lets us write any physically
reasonable wave function as a sum over Gaussians. Provided we are dealing only
with linear operations – the case throughout here – we can work directly with
Gaussian test functions. By Morlet wavelet analysis the results will then be
valid for any physically reasonable wave functions.

###  2.5 Lagrangian

To sum over the paths – to construct the path integral – we will need to
weight each path by the exponential of the action, where the action is defined
as the integral of the Lagrangian over the laboratory time:

$e^{\imath\int\limits_{0}^{T}d\tau L\left(x^{\mu},\dot{x}^{\mu}\right)}$ (27)

We require a Lagrangian which:

  1. 1.

Is manifestly covariant,

  2. 2.

Produces the correct classical equations of motion,

  3. 3.

And gives the correct Schrödinger equation.

We would further prefer a Lagrangian which is the same for both SQM and TQM.
This will let us argue that we are treating SQM and TQM with the most complete
possible equality.

Somewhat surprisingly such a Lagrangian exists. In Goldstein’s well-known text
on classical mechanics [63] we find:

$L\left(x^{\mu},\dot{x}^{\mu}\right)=-\frac{1}{2}m\dot{x}^{\mu}\dot{x}_{\mu}-q\dot{x}^{\mu}A_{\mu}\left(x\right)$
(28)

The potentials are not themselves functions of the laboratory time $\tau$. The
mass $m$ is the rest mass of the particle, an invariant.

This Lagrangian is unusual in that it uses four independent variables (the
usual three space coordinates plus a time variable) but still gives the
familiar classical equations of motion (see B).

This Lagrangian therefore provides a natural bridge from a three to a four
dimensional picture.

The classical equations of motion are still produced if we add a dimensionless
scale $a$ and an additive constant $b$ to the Lagrangian:

$-\frac{1}{2}am\dot{x}^{\mu}\dot{x}_{\mu}-aq\dot{x}^{\mu}A_{\mu}\left(x\right)-ab\frac{m}{2}$
(29)

The Lagrangian is therefore only determined up to $a$ and $b$. The requirement
that we match the SQM results will fix $a$ and $b$ (subsection 3.3).

###  2.6 Convergence

How do we get the integrals over the paths to converge without breaking
covariance?

We compute the path integral for the kernel by slicing the clock time into an
infinite number of intervals and integrating over each:

$K_{BA}=\lim_{N\to\infty}C_{N}\int\prod\limits_{n=1}^{N-1}dt_{n}d\vec{x}_{n}e^{\imath\varepsilon\sum\limits_{j=1}^{N}L_{j}}$
(30)

with $C_{N}$ an appropriate normalization factor.

Consider the discrete form of the Lagrangian. We use a tilde to mark the
coordinate time part and an overbar to mark the space part:

$L_{j}\equiv\tilde{L}_{j}^{t}+\bar{L}_{j}^{\vec{x}}+L_{j}^{m}$ (31)
$\tilde{L}_{j}^{t}\equiv-a\frac{m}{2}\left(\frac{t_{j}-t_{j-1}}{\varepsilon}\right)^{2}-qa\frac{t_{j}-t_{j-1}}{\varepsilon}\frac{\Phi\left(x_{j}\right)+\Phi\left(x_{j-1}\right)}{2}$
(32) $\bar{L}_{j}^{\vec{x}}\equiv
a\frac{m}{2}\left(\frac{\vec{x}_{j}-\vec{x}_{j-1}}{\varepsilon}\right)^{2}+qa\frac{\vec{x}_{j}-\vec{x}_{j-1}}{\varepsilon}\cdot\frac{\vec{A}\left(x_{j}\right)+\vec{A}\left(x_{j-1}\right)}{2}$
(33) $L_{j}^{m}\equiv-ab\frac{m}{2}$ (34)

We are using the mid-point rule, averaging the scalar and the vector
potentials over the start and end points of the step, by analogy with the rule
for three dimensions (Schulman [44], Grosche and Steiner [49]).

Now look at a single step for the free case, vector potential $A_{\mu}$ zero:

$\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}dt_{j}d\vec{x}e^{-\imath
am\frac{\left(t_{j}-t_{j-1}\right)^{2}}{2\varepsilon}+\imath
am\frac{\left(\vec{x}_{j}-\vec{x}_{j-1}\right)^{2}}{2\varepsilon}}$ (35)

The formal tricks normally used to ensure convergence do not work here (e.g.
Kashiwa or Zinn-Justin [48, 50]). Perhaps the most popular of these is the use
of Wick rotation to shift to a Euclidean time:

$\tau\to-\imath\tau$ (36)

This causes integrals to converge rapidly going into the future, but makes the
past inaccessible. For instance, factors of
$\exp\left({-\imath\omega\tau}\right)$ – which spring up everywhere in path
integrals – converge going into the future, but blow up going into the past.
If we are to treat time on the same footing as space – our central assumption
– then past and future must be treated as symmetrically as left and right.

Another approach is to add a small convergence factor at a cleverly chosen
spot in the arguments of the exponentials. But if we attach a convergence
factor to $t$ and $x$ separately, we break manifest covariance. If we attach
our convergence factor to both, the fact that the $t$ and $x$ parts enter with
opposite sign means any convergence factor that works for one will fail for
the other. We could try attaching one to the mass $m$, but this also fails.
For instance if $a>0$ and we subtract a small factor of $\imath\delta$ from
the mass:

$m\to m-\imath\delta$ (37)

the $t$ integral converges but the $x$ integral diverges.

We recall the kernel has meaning only when applied to a specific physical wave
function. If we break the incoming wave up into Morlet wavelets and then into
Gaussian test functions, we see that each integral converges by inspection,
the factor $e^{-\frac{1}{2}\left(\frac{t-l}{s}\right)^{2}}$ ensures this.

So for physically significant wave functions, there is no problem in the first
place. Effectively we are taking seriously the point that the path integral
kernel is a distribution, only meaningful with respect to specific wave
functions.

###  2.7 Normalization

Now that we have our path integrals converging, we have to normalize them. If
we start from the Schrödinger equation, the normalization is wired in. But in
path integrals we are a bit at sea.

We will here deal with the free case, verifying the normalization is correct
in the general case in C.

The normalization factor for $N$ steps we will call $C_{N}$. The defining
requirement is that, if the initial wave function is normalized to one, then
with the inclusion of $C_{N}$, the final wave function will be normalized to
one as well:

$\int{d{t_{0}}d{{\vec{x}}_{0}}\left|{{\psi_{0}}{{\left({{t_{0}},{{\vec{x}}_{0}}}\right)}^{2}}}\right|=1}\to\int{d{t_{N}}d{{\vec{x}}_{N}}\left|{{\psi_{N}}{{\left({{t_{N}},{{\vec{x}}_{N}}}\right)}^{2}}}\right|=1}$
(38)

If $C_{N}$ depends on the particular $\psi_{0}$, then we have failed.

We now compute the factor of $C_{N}$.

####  2.7.1 Normalization in time

We start with the coordinate time dimension only. Consider a Gaussian test
function centered on an initial position in coordinate time $\bar{t}_{0}$:

$\tilde{\varphi}_{0}\left(t_{0}\right)\equiv\sqrt[4]{\frac{1}{\pi\sigma_{t}^{2}}}e^{-\imath
E_{0}t_{0}-\frac{\left(t_{0}-\bar{t}_{0}\right)^{2}}{2\sigma_{t}^{2}}}$ (39)

We write the kernel for the time part as:

$\tilde{K}_{\tau}\left(t_{N};t_{0}\right)\sim\int\prod\limits_{j=1}^{N-1}dt_{j}e^{-\imath
am\sum\limits_{k=1}^{N}\frac{\left(t_{k}-t_{k-1}\right)^{2}}{2\varepsilon}}$
(40)

The wave function after the initial integral over $t_{0}$ is:

$\tilde{\varphi}_{\varepsilon}\left(t_{1}\right)=\int
dt_{0}e^{-\imath\frac{am}{2\varepsilon}\left(t_{1}-t_{0}\right)^{2}}\tilde{\varphi}_{0}\left(t_{0}\right)$
(41)

or:

${\tilde{\varphi}}_{\varepsilon}\left(t_{1}\right)=\sqrt{\frac{2\pi\varepsilon}{\imath
am}}\sqrt[4]{\frac{1}{\pi\sigma_{t}^{2}}}\sqrt{\frac{1}{f_{\varepsilon}^{\left(t\right)}}}e^{-\imath
E_{0}t_{1}+\imath\frac{E_{0}^{2}}{2am}\varepsilon-\frac{1}{2\sigma_{t}^{2}f_{\varepsilon}^{\left(t\right)}}\left(t_{1}-\bar{t}_{0}-\frac{E_{0}}{am}\varepsilon\right)^{2}}$
(42)

with the dispersion factor $f_{\tau}^{\left(t\right)}\equiv
1-\imath\frac{\tau}{am\sigma_{t}^{2}}$.

The normalization requirement is:

$1=\int
dt_{1}\tilde{\varphi}_{\varepsilon}^{*}\left(t_{1}\right)\tilde{\varphi}_{\varepsilon}\left(t_{1}\right)$
(43)

The first step normalization is correct if we multiply the kernel by a factor
of $\sqrt{\frac{\imath am}{2\pi\varepsilon}}$. Since this normalization factor
does not depend on the laboratory time the overall normalization for $N$
infinitesimal kernels is the product of $N$ of these factors:

$C_{N}\equiv\sqrt{\frac{\imath am}{2\pi\varepsilon}}^{N}$ (44)

Note also that the normalization does not depend on the specifics of the
Gaussian test function (the values of $E_{0}$, $\sigma_{t}^{2}$, and
$\bar{t}_{0}$) so it is valid for an arbitrary sum of Gaussian test functions
as well. And therefore, by Morlet wavelet decomposition, for an arbitrary wave
function.

As noted, the phase is arbitrary. If we were working the other way, from
Schrödinger equation to path integral, the phase would be determined by the
Schrödinger equation itself. The specific phase choice we are making here has
been chosen to help ensure the four dimensional Schrödinger equation is
manifestly covariant, see below. We may think of the phase choice as a choice
of gauge (see D).

Therefore the expression for the free kernel in coordinate time is (with
$t^{\prime\prime}\equiv t_{N}$, $t^{\prime}\equiv t_{0}$):

$\tilde{K}_{\tau}\left(t^{\prime\prime};t^{\prime}\right)=\int
dt_{1}dt_{2}\ldots dt_{N-1}\sqrt{\frac{\imath
am}{2\pi\varepsilon}}^{N}e^{-\imath\sum\limits_{j=1}^{N}\left(\frac{am}{2\varepsilon}\left(t_{j}-t_{j-1}\right)^{2}\right)}$
(45)

Doing the integrals we get:

$\tilde{K}_{\tau}\left(t^{\prime\prime};t^{\prime}\right)=\sqrt{\frac{\imath
am}{2\pi\tau}}e^{-\imath
am\frac{\left(t^{\prime\prime}-t^{\prime}\right)^{2}}{2\tau}}$ (46)

and free wave functions in coordinate time:

${{\tilde{\varphi}}_{\tau}}\left(t\right)=\sqrt[4]{{\frac{1}{{\pi\sigma_{t}^{2}}}}}\sqrt{\frac{1}{{f_{\tau}^{\left(t\right)}}}}{e^{-\imath{E_{0}}t-\frac{1}{{2\sigma_{t}^{2}f_{\tau}^{\left(t\right)}}}{{\left({t-{{\bar{t}}_{0}}-\frac{{E_{0}}}{am}\tau}\right)}^{2}}+\imath\frac{{E_{0}^{2}}}{{2am}}\tau}}$
(47)

####  2.7.2 Normalization in space

We redo the analysis for coordinate time for space. We use the
correspondences:

$t\to x,m\to-m,\bar{t}_{0}\to\bar{x}_{0},E_{0}\to-
p_{0},\sigma_{t}^{2}\to\sigma_{x}^{2}$ (48)

With these we can write down the equivalent set of results by inspection.
Since we will need the results below, we do this explicitly. We get the
initial Gaussian test function:

$\bar{\varphi}_{0}\left(x_{0}\right)=\sqrt[4]{\frac{1}{\pi\sigma_{x}^{2}}}e^{\imath
p_{0}x_{0}-\frac{\left(x_{0}-\bar{x}_{0}\right)^{2}}{2\sigma_{x}^{2}}}$ (49)

free kernel:

$\bar{K}_{\tau}\left(x^{\prime\prime};x^{\prime}\right)\sim\int
dx_{1}dx_{2}\ldots
dx_{N-1}e^{\imath\sum\limits_{j=1}^{N}\frac{am}{2\varepsilon}\left(x_{j}-x_{j-1}\right)^{2}}$
(50)

and normalized kernel:

$\bar{K}_{\tau}\left(x^{\prime\prime};x^{\prime}\right)=\sqrt{-\frac{\imath
am}{2\pi\tau}}e^{\imath
am\frac{\left(x^{\prime\prime}-x^{\prime}\right)^{2}}{2\tau}}$ (51)

The kernel matches the usual (non-relativistic) kernel [64, 44] if $a=1$.

The wave function is:

$\bar{\varphi}_{\tau}\left(x\right)=\sqrt[4]{\frac{1}{\pi\sigma_{x}^{2}}}\sqrt{\frac{1}{f_{\tau}^{\left(x\right)}}}e^{\imath
p_{0}x-\frac{1}{2\sigma_{x}^{2}f_{\tau}^{\left(x\right)}}\left(x-\bar{x}_{0}-\frac{p_{0}}{am}\tau\right)^{2}-\imath\frac{p_{0}^{2}}{2am}\tau}$
(52)

with the definition of the dispersion factor
$f_{\tau}^{\left(x\right)}=1+\imath\frac{\tau}{am\sigma_{x}^{2}}$ parallel to
that for coordinate time (but with opposite sign for the imaginary part).

####  2.7.3 Normalization in time and space

The full kernel is the product of the coordinate time kernel, the three space
kernels, and the constant term $e^{-\imath\frac{abm}{2}\tau}$. We understand
$x$ to refer to coordinate time and all three space dimensions:

$K_{\tau}\left(x^{\prime\prime};x^{\prime}\right)=-\imath\frac{a^{2}m^{2}}{4\pi^{2}\tau^{2}}e^{-\frac{\imath
am}{2\tau}\left(x^{\prime\prime}-x^{\prime}\right)^{2}-\imath\frac{abm}{2}\tau}$
(53)

We have done the analysis only for Gaussian test functions, but by Morlet
wavelet decomposition it is completely general.

###  2.8 Formal expression

We now have the full path integral:

$K_{\tau}\left(x^{\prime\prime};x^{\prime}\right)=\lim\limits_{N\to\infty}\int\mathcal{D}xe^{\imath\sum\limits_{j=1}^{N}L_{j}\epsilon}$
(54)

with the measure:

$\mathcal{D}x\equiv\left(-\imath
a^{2}\frac{m^{2}}{4\pi^{2}\varepsilon^{2}}\right)^{N}\prod\limits_{n=1}^{N-1}d^{4}x_{n}$
(55)

and the discretized Lagrangian at each step:

$L_{j}\equiv-
am\frac{\left(x_{j}-x_{j-1}\right)^{2}}{2\varepsilon^{2}}-aq\frac{x_{j}-x_{j-1}}{\epsilon}\frac{A\left(x_{j}\right)+A\left(x_{j-1}\right)}{2}-ab\frac{m}{2}$
(56)

We will show that $a=b=1$ in the next section.

##  3 Schrödinger equation

###  3.1 Overview

The path integral and Schrödinger equation views are complementary. We need
both to fully understand either.

We derive the Schrödinger equation from the path integral by taking the short
time limit of the path integral form.

By comparing the result to the Klein-Gordon equation – and making a reasonable
assumption about the long time evolution of the wave functions – we are able
to fix the additive and scale constants in the Lagrangian.

The resulting equation looks like the Klein-Gordon equation over short times
but shows some drift over longer times. We use some heuristic arguments to
estimate the scale of the long term drift as of order picoseconds, a million
times longer than the attosecond scale of the time dispersions we are
primarily concerned with here. We will therefore be able to largely ignore
this drift.

With $a,b$ defined, we look at a further problem. We have done the derivation
of path integral and Schrödinger equation forms from Alice’s perspective. But
what of Bob, jetting around like a fusion powered mosquito?

We resolve this conflict by arguing that we can find a natural rest frame that
both can use. Starting with an argument of Weinberg’s, we argue we can
associate an energy-momentum tensor with spacetime. This means we can
associate a local rest frame with spacetime. And this local rest frame can
provide the neutral and agreed defining frame for TQM.

This will complete the formal development of TQM.

Before turning to applications, we will then look at the relationship of TQM
to the relativistic dynamics program. We will argue that TQM may be understood
as a member of that program aggressively specialized to achieve
falsifiability.

###  3.2 Derivation of the Schrödinger equation

Normally the path integral expression is derived from the Schrödinger
equation. But because for us the path integral provides the defining
formulation we need to run the analysis in the “wrong” direction.

Our starting point is a derivation of the path integral from the Schrödinger
equation by Schulman [44]. We run his derivation in reverse and with one extra
dimension222This derivation is done for the free case in Fanchi [36]. .

We start with the discrete form of the path integral. We consider a single
step of length $\epsilon$, taking $\epsilon\to 0$ at the end. Because of this,
only terms first order in $\epsilon$ are needed.

Following Schulman, we define the coordinate difference:

$\xi\equiv x_{j}-x_{j+1}$ (57)

We rewrite the functions of $x_{j}$ as functions of $\xi$ and $x_{j+1}$. We
expand the vector potential:

$A_{\nu}\left(x_{j}\right)=A_{\nu}\left(x_{j+1}\right)+\left(\xi^{\mu}\partial_{\mu}\right)A_{\nu}\left(x_{j+1}\right)+\ldots$
(58)

and the wave function:

$\psi_{\tau}\left(x_{j}\right)=\psi_{\tau}\left(x_{j+1}\right)+\left(\xi^{\mu}\partial_{\mu}\right)\psi_{\tau}\left(x_{j+1}\right)+\frac{1}{2}\xi^{\mu}\xi^{\nu}\partial_{\mu}\partial_{\nu}\psi_{\tau}\left(x_{j+1}\right)+\ldots$
(59)

giving:

$\begin{array}[]{c}\psi_{\tau+\varepsilon}\left(x_{j+1}\right)=\sqrt{\frac{\imath
am}{2\pi\varepsilon}}\sqrt{-\frac{\imath am}{2\pi\varepsilon}}^{3}\int
d^{4}\xi e^{-\frac{\imath am\xi^{2}}{2\varepsilon}-\imath
ab\frac{m}{2}\varepsilon}\\\ \times e^{\imath
aq\xi^{\nu}\left(A_{\nu}\left(x_{j+1}\right)+\frac{1}{2}\left(\xi^{\mu}\partial_{\mu}\right)A_{\nu}\left(x_{j+1}\right)+\ldots\right)}\\\
\times\left(\psi_{\tau}\left(x_{j+1}\right)+\left(\xi^{\mu}\partial_{\mu}\right)\psi_{\tau}\left(x_{j+1}\right)+\frac{1}{2}\xi^{\mu}\xi^{\nu}\partial_{\mu}\partial_{\nu}\psi_{\tau}\left(x_{j+1}\right)+\ldots\right)\end{array}$
(60)

We now expand in powers of $\xi\sim\sqrt{\epsilon}$. We do not need more than
the second power:

$\begin{array}[]{c}\psi_{\tau+\varepsilon}\left(x_{j+1}\right)=\sqrt{\frac{\imath
am}{2\pi\varepsilon}}\sqrt{-\frac{\imath am}{2\pi\varepsilon}}^{3}\int
d^{4}\xi e^{-\frac{\imath am\xi^{2}}{2\varepsilon}}\\\ \times\left(1+\imath
aq\xi^{\nu}A_{\nu}\left(x_{j+1}\right)+\frac{\imath
aq}{2}\xi^{\nu}\xi^{\mu}\partial_{\mu}A_{\nu}\left(x_{j+1}\right)-\frac{a^{2}q^{2}}{2}\xi^{\mu}A_{\mu}\left(x_{j+1}\right)\xi^{\nu}A_{\nu}\left(x_{j+1}\right)-\imath\frac{abm\varepsilon}{2}\right)\\\
\times\left(\psi_{\tau}\left(x_{j+1}\right)+\left(\xi^{\mu}\partial_{\mu}\right)\psi_{\tau}\left(x_{j+1}\right)+\frac{1}{2}\xi^{\mu}\xi^{\nu}\partial_{\mu}\partial_{\nu}\psi_{\tau}\left(x_{j+1}\right)+\ldots\right)\end{array}$
(61)

The term zeroth order in $\xi$ gives:

$\sqrt{\frac{\imath am}{2\pi\varepsilon}}\sqrt{-\frac{\imath
am}{2\pi\varepsilon}}^{3}\int d\xi^{4}e^{-\frac{\imath
am\xi^{2}}{2\varepsilon}}=1$ (62)

This is not surprising; the normalization above was chosen to do this.

Terms linear in $\xi$ give zero when integrated.

The terms second order in $\xi$ (first in $\epsilon$) are:

$\left(\begin{array}[]{c}-\imath\frac{abm\varepsilon}{2}+\imath
aq\left(\xi^{\nu}A_{\nu}\right)\left(\xi^{\mu}\partial_{\mu}\right)+\frac{\imath
aq}{2}\xi^{\nu}\xi^{\mu}\left(\partial_{\mu}A_{\nu}\right)\hfill\\\
-\frac{a^{2}q^{2}}{2}\left(\xi^{\mu}A_{\mu}\right)\left(\xi^{\nu}A_{\nu}\right)+\frac{1}{2}\xi^{\mu}\xi^{\nu}\partial_{\mu}\partial_{\nu}\hfill\end{array}\right){\psi_{\tau}}$
(63)

Integrals over off-diagonal powers of order $\xi^{2}$ give zero. Integrals
over diagonal $\xi^{2}$ terms give:

$\begin{array}[]{c}\sqrt{\frac{\imath am}{2\pi\varepsilon}}\int
d\xi_{0}e^{-\frac{\imath
am\xi_{0}^{2}}{2\varepsilon}}\xi_{0}^{2}=\frac{\varepsilon}{\imath am}\\\
\sqrt{-\frac{\imath am}{2\pi\varepsilon}}\int d\xi_{i}e^{\frac{\imath
am\xi_{i}^{2}}{2\varepsilon}}\xi_{i}^{2}=-\frac{\varepsilon}{\imath
am}\end{array}$ (64)

The expression for the wave function is therefore:

$\psi_{\tau+\varepsilon}=\psi_{\tau}-\frac{\imath
abm\varepsilon}{2}\psi_{\tau}+\frac{q\varepsilon}{m}\left(A^{\mu}\partial_{\mu}\right)\psi_{\tau}+\frac{q}{2m}\varepsilon\left(\partial^{\mu}A_{\mu}\right)\psi_{\tau}+\frac{\imath
aq^{2}\varepsilon}{2m}A^{2}\psi_{\tau}-\frac{\imath\varepsilon}{2am}\partial^{2}\psi_{\tau}$
(65)

Taking the limit $\epsilon\to 0$ and multiplying by $\imath$, we get the
Schrödinger equation for TQM:

$\imath\frac{\partial\psi_{\tau}}{\partial\tau}=ab\frac{m}{2}\psi_{\tau}+\frac{\imath
q}{m}\left(A^{\mu}\partial_{\mu}\right)\psi_{\tau}+\frac{\imath
q}{2m}\left(\partial^{\mu}A_{\mu}\right)\psi_{\tau}-\frac{aq^{2}}{2m}A^{2}\psi_{\tau}+\frac{1}{2ma}\partial^{2}\psi_{\tau}$
(66)

or:

$\imath\frac{\partial\psi_{\tau}}{\partial\tau}\left(t,\vec{x}\right)=-\frac{1}{2ma}\left(\left(\imath\partial_{\mu}-aqA_{\mu}\left(t,\vec{x}\right)\right)\left(\imath\partial^{\mu}-aqA^{\mu}\left(t,\vec{x}\right)\right)-a^{2}bm^{2}\right)\psi_{\tau}\left(t,\vec{x}\right)$
(67)

If we make the customary identifications $\imath\frac{\partial}{\partial t}\to
E$, $-\imath\vec{\nabla}\to\hat{\vec{p}}$ or $\imath\partial_{\mu}\to p_{\mu}$
we have:

$\imath\frac{\partial\psi_{\tau}}{\partial\tau}=-\frac{1}{2ma}\left(\left(p_{\mu}-aqA_{\mu}\right)\left(p^{\mu}-aqA^{\mu}\right)-a^{2}bm^{2}\right)\psi_{\tau}$
(68)

###  3.3 Long, slow approximation

We can now fix the scale and additive constants by looking at the behavior of
the Schrödinger equation over longer times.

In his development of quantum mechanics from a time-dependent perspective
[65], Tannor used a requirement of constructive interference in time to derive
the Bohr condition for the allowed atomic orbitals. We use a similar approach
here.

If we average over a sufficiently long period of time, the results will be
dominated by the components with:

$\imath\frac{\partial\psi_{\tau}\left(x\right)}{\partial\tau}=0$ (69)

The argument here is not that the typical variation from the long, slow
solution is small, but rather that over time interactions with the system in
question will tend to be dominated by interactions with the stabler, slower
moving components. Interactions with more rapidly varying components will tend
to average to zero.

Accepting this, then the right side looks like the Klein-Gordon equation. To
complete this identification, first look at the case with the vector potential
$A$ zero:

$\left(\hat{p}^{2}-a^{2}bm^{2}\right)\psi=0\to a^{2}b=1$ (70)

Now when $A$ is not zero we have:

$\left(\left(\hat{p}-aqA\right)^{2}-m^{2}\right)\psi=0\to a=1\to b=1$ (71)

We will refer to this as the “long, slow approximation”.

In the free case, the long, slow approximation picks out the on-shell
components:

$\left(\hat{p}^{2}-m^{2}\right)\psi=0$ (72)

And more generally the solutions of the Klein-Gordon equation with the minimal
substitution $p\to p-qA$:

$\left(\left(\hat{p}-qA\right)^{2}-m^{2}\right)\psi=0$ (73)

The two constants are now fixed. The full Schrödinger equation is:

$\imath\frac{\partial\psi_{\tau}}{\partial\tau}\left(t,\vec{x}\right)=-\frac{1}{2m}\left(\left(\imath\partial_{\mu}-qA_{\mu}\left(t,\vec{x}\right)\right)\left(\imath\partial^{\mu}-qA^{\mu}\left(t,\vec{x}\right)\right)-m^{2}\right)\psi_{\tau}\left(t,\vec{x}\right)$
(74)

and in momentum space:

$\imath\frac{\partial\psi_{\tau}}{\partial\tau}=-\frac{1}{{2m}}\left({\left({{p_{\mu}}-q{A_{\mu}}}\right)\left({{p^{\mu}}-q{A^{\mu}}}\right)-m^{2}}\right){\psi_{\tau}}$
(75)

The free Schrödinger equation is:

$2m\imath\frac{\partial\psi_{\tau}}{\partial\tau}\left({t,\vec{x}}\right)=\left({{\partial_{\mu}}{\partial^{\mu}}+{m^{2}}}\right){\psi_{\tau}}\left({t,\vec{x}}\right)$
(76)

and in momentum space:

$2m\imath\frac{\partial\psi_{\tau}}{\partial\tau}=-\left({{p_{\mu}}{p^{\mu}}-{m^{2}}}\right){\psi_{\tau}}\left({t,\vec{x}}\right)$
(77)

We establish in C this equation is unitary. Therefore if a wave function is
normalized at $\tau=0$:

$1=\int{dtd\vec{x}\psi_{0}^{*}\left({t,\vec{x}}\right){\psi_{0}}\left({t,\vec{x}}\right)}$
(78)

then at any later clock time $\tau>0$ we will have:

$1=\int{dtd\vec{x}\psi_{\tau}^{*}\left({t,\vec{x}}\right){\psi_{\tau}}\left({t,\vec{x}}\right)}$
(79)

We next estimate the time scales over which we expect the long, slow
approximation to be valid.

###  3.4 How long and how slow?

We have argued that we can fix the scaling and additive constants by looking
at the behavior of the Schrödinger equation over long times. What do we mean
by long times?

To see the relevant scale, we estimate the clock frequency $f$:

$f\sim-\frac{{{E^{2}}-{{\vec{p}}^{2}}-{m^{2}}}}{{2m}}$ (80)

In the non-relativistic case $E$ is of order mass plus kinetic energy:

$E\sim m+\frac{{{\vec{p}}^{2}}}{{2m}}$ (81)

So we have:

${E^{2}}-{{\vec{p}}^{2}}-{m^{2}}\sim{\left({m+\frac{{{\vec{p}}^{2}}}{{2m}}}\right)^{2}}-{{\vec{p}}^{2}}-{m^{2}}={\left({\frac{{{\vec{p}}^{2}}}{{2m}}}\right)^{2}}$
(82)

This is just the kinetic energy, squared. In an atom the kinetic energy is of
order the binding energy:

$\frac{{{\vec{p}}^{2}}}{{2m}}\sim eV$ (83)

So the numerator is of order $eV$ squared. But the denominator is of order
$MeV$. So we can estimate the clock frequency $f$ as:

$f\sim\frac{{e{V^{2}}}}{{MeV}}\sim{10^{-6}}eV$ (84)

Energies of millionths of an electron volt ${10^{-6}eV}$ correspond to times
of order millions of attoseconds $10^{6}as$ or picoseconds, a million times
longer than the natural time scale of the effects we are looking at. So the
long, slow approximation is reasonable.

The physical picture that emerges is of a particle that is extended in time on
the same basis as space, with its wave function extend in all four dimensions,
and – if we think in terms of path integrals – the associated paths wandering
around in all four dimensions, not just the traditional three.

It may be amusing to note that from this perspective there is no such thing as
an onshell particle; in momentum space the onshell part of the wave function
is always a set of measure zero. It is only over scales of picoseconds and
greater that an onshell description of the particle may give an acceptable
approximation.

###  3.5 Observer independent choice of frame

There remains one piece of the puzzle; we need to establish that the use of
the clock time $\tau$ does not itself violate covariance.

If we did the above derivation for Bob rather than Alice, we would see Alice’s
clock time replaced by Bob’s clock time $\tau\to\gamma\tau$
($\gamma\equiv\frac{1}{{\sqrt{1-{v^{2}}}}}$) where $v$ is his velocity
relative to her.

We therefore have one free parameter left to fix before we can declare our
analysis free of free parameters.

If Bob is not going that quickly (relative to Alice) the errors created by
ambiguities with respect to $\tau$ will introduce only small corrections; of
only second order and therefore not relevant for falsifiability.

Even if we were prepared to accept that, establishing frame independence is
interesting as a point of principle.

This may be done in a natural way by making use of an observation from
Weinberg [66]. Per Weinberg, we may treat the Einstein field equation for
general relativity as representing conservation of energy-momentum when
exchanges of energy momentum with spacetime are included.

Consider the Einstein field equations:

${G_{\mu\nu}}\equiv{R_{\mu\nu}}-\frac{1}{2}{g_{\mu\nu}}R=-8\pi G{T_{\mu\nu}}$
(85)

Rewrite as:

${\left({{G^{\mu\nu}}+8\pi G{T^{\mu\nu}}}\right)_{;\nu}}=0$ (86)

We may use this to associate an energy momentum tensor ($t_{\mu\nu}$ in
Weinberg’s notation) with local space time. Define:

${g_{\mu\nu}}={\eta_{\mu\nu}}+{h_{\mu\nu}}$ (87)

where $h_{\mu\nu}$ vanishes at infinity but is not assumed small. The part of
the Ricci tensor linear in $h$ is:

$R_{\mu\nu}^{\left(1\right)}\equiv\frac{1}{2}\left({\frac{{{\partial^{2}}h_{\lambda}^{\lambda}}}{{\partial{x^{\mu}}\partial{x^{\nu}}}}-\frac{{{\partial^{2}}h_{\mu}^{\lambda}}}{{\partial{x^{\lambda}}\partial{x^{\nu}}}}-\frac{{{\partial^{2}}h_{\nu}^{\lambda}}}{{\partial{x^{\lambda}}\partial{x^{\mu}}}}+\frac{{{\partial^{2}}{h_{\mu\nu}}}}{{\partial{x^{\lambda}}\partial{x_{\lambda}}}}}\right)$
(88)

The exact Einstein equations may be written as:

$R_{\mu\nu}^{\left(1\right)}-\frac{1}{2}{\eta_{\mu\nu}}R_{\lambda}^{\left(1\right)\lambda}=-8\pi
G\left({{T_{\mu\nu}}+{t_{\mu\nu}}}\right)$ (89)

where $t_{\mu\nu}$ is defined by:

${t_{\mu\nu}}\equiv\frac{1}{{8\pi
G}}\left({{R_{\mu\nu}}-\frac{1}{2}{g_{\mu\nu}}R_{\lambda}^{\lambda}-R_{\mu\nu}^{\left(1\right)}+\frac{1}{2}{\eta_{\mu\nu}}R_{\lambda}^{\left(1\right)\lambda}}\right)$
(90)

Weinberg then argues we may interpret $t_{\mu\nu}$ as the energy-momentum of
the gravitational field itself.

Accepting this, we change to a coordinate frame in which $t_{\mu\nu}$ is
diagonalized. We will refer to this as the rest frame of the vacuum or the $V$
frame. We can treat this $V$ frame as the defining frame for the four
dimensional Schrödinger equation. As the $V$ frame is invariant (up to
rotations in three-space) we now have an invariant definition of the four
dimensional Schrödinger equation.

This is obviously going to be a free-falling frame. So Alice and Bob – if they
are working in a terrestrial laboratory – will have to adjust their
calculations to include a correction for the upwards force the laboratory
floor exerts against them. If their colleague Vera is working in an orbiting
laboratory, she will be able to calculate without correction.

Here after, unless stated to contrary, we will assume we are working in the
“rest frame of the vacuum”.

###  3.6 Relationship between TQM and the Relativistic Dynamics program

> “In other words, we are trying to prove ourselves wrong as quickly as
> possible, because only in that way can we find progress.” – Richard P.
> Feynman [2]

We have now fully defined the path integral expression and the Schrödinger
equation.

It is appropriate therefore to pause to look at relationship of TQM to the
relativistic dynamics program. (A helpful overview of the relativistic
dynamics program is provided in Fanchi [35]).

####  3.6.1 TQM as part of the Relativistic Dynamics program

We start with Feynman [17], who uses the Lagrangian:

${L_{Feynman}}\left({{x^{\mu}},{{\dot{x}}^{\mu}}}\right)=-{{\dot{x}}^{\mu}}{{\dot{x}}_{\mu}}-q{{\dot{x}}^{\mu}}{A_{\mu}}\left(x\right)$
(91)

giving:

$\imath\frac{\partial}{{\partial
s}}{\psi_{s}}=\frac{{\left({{p^{\mu}}-{q}{A^{\mu}}}\right)\left({{p_{\mu}}-{q}{A_{\mu}}}\right)}}{2}{\psi_{s}}$
(92)

However with this approach, as Feynman notes, each particle would need its own
evolution parameter $s$. This would make extension to the multiple particle
case problematic.

There are many variations on this in the literature. Fanchi [36] derives:

$\imath\frac{\partial}{{\partial
s}}{\psi_{s}}=\frac{{\left({{p^{\mu}}-{q}{A^{\mu}}}\right)\left({{p_{\mu}}-{q}{A_{\mu}}}\right)}}{2m}{\psi_{s}}$
(93)

This differs from Feynman’s by a factor of $\frac{1}{m}$. Fanchi’s evolution
parameter $s$ is for all practical purposes the negative of our clock time
$\tau$.

Land and Horwitz [37] give:

$\left({\imath\frac{\partial}{{\partial
s}}+{q}{a_{5}}}\right){\psi_{s}}=\frac{{\left({{p^{\mu}}-{q}{A^{\mu}}}\right)\left({{p_{\mu}}-{q}{A_{\mu}}}\right)}}{{2m}}{\psi_{s}}$
(94)

This is Fanchi’s with the addition of the $a_{5}$, a gauge term.

If we take $s\to-\tau$ and make the choice of gauge (see D):

${a_{5}}\Rightarrow\frac{{m}}{{2q}}$ (95)

we get:

$\left({\imath\frac{\partial}{{\partial\tau}}}\right){\psi_{s}}=-\frac{{\left({{p^{\mu}}-q{A^{\mu}}}\right)\left({{p_{\mu}}-q{A_{\mu}}}\right)-{m^{2}}}}{{2m}}{\psi_{s}}$
(96)

which is the same as our Schrödinger equation (equation 74).

At this point we may argue that as the right hand-side is the Klein-Gordon
equation, and the Klein-Gordon equation is strongly confirmed by experimental
evidence we have:

$0\approx\frac{{\left({{p^{\mu}}-q{A^{\mu}}}\right)\left({{p_{\mu}}-q{A_{\mu}}}\right)-{m^{2}}}}{{2m}}{\psi_{s}}$
(97)

So the combination of the mass gauge ($a_{5}=\frac{m}{2q}$) and the comparison
with the Klein-Gordon equation give us the long slow approximation.

We have therefore placed our Schrödinger equation within the context of the
relativistic program. This will let us use results from that program in the
appropriate limit.

####  3.6.2 TQM and falsifiability

TQM may be thought of as a specialization of the relativistic dynamics program
aimed at making the hypothesis “time should be treated on the same basis as
space in quantum mechanics” falsifiable. This means in turn that we have to be
able to rule out the class of such theories, not just one in particular.

To do this we need to extend quantum mechanics to include time in a way that
depends only on covariance. We can admit no free or adjustable parameters; any
such risk falsifiability.

We may understand each step of the program here from this point of view.

  1. 1.

We used clock time as the “evolution parameter”. This is defined operationally
– by the laboratory clock – and therefore introduces no free parameters.

  2. 2.

We used path integrals as the defining representation; with these the only
change we need to make is to have the paths extend in four rather than three
dimensions.

  3. 3.

We could not use plane waves or delta functions as the initial wave functions:
as they are not physical, they introduce the risk of inducing mathematical
artifacts and again threatening falsifiability.

  4. 4.

We needed to treat general wave functions, not just a specialized subset.
Otherwise the results might be conditional on the subset chosen.

  5. 5.

We needed to ensure that our path integrals are convergent without the use of
tricks. Otherwise the tricks can bring the results into question. The use of
Morlet wavelet decomposition solved this and the previous two problems.

  6. 6.

The resulting Schrödinger equation depends on the clock time. To eliminate
such dependence (to lowest order) we showed that the time scales associated
with the clock time are of order picoseconds, while the scales associated with
dispersion in time are of order attoseconds – a million times smaller. So long
as we work at the attosecond scale we should be able to ignore effects
associated with clock time.

  7. 7.

The clock time in turn depends on the choice of the laboratory frame. To
further control for effects of the specific choice of laboratory frame, we
showed we can define an invariant frame (the “rest frame of the vacuum”). By
defining all calculations with reference to this invariant frame we eliminate
the dependence of the clock time on the choice of a specific laboratory frame.

While we have fully defined the path integral expression and the Schrödinger
equation, there are still some further questions that need to be addressed if
we are to achieve falsifiability:

  1. 1.

How do we estimate the initial wave functions in time in a robust and frame
independent way (subsection 4.1)?

  2. 2.

How do we understand the evolution of the wave function with respect to clock
time: with the long, slow approximation we appear to be saying the wave
function is frozen in time, that
$\imath\frac{\partial}{{\partial\tau}}\psi\approx 0$, which is clearly far
from the case (subsection 4.2)?

  3. 3.

How do we define the rules for detection in a way which is manifestly
covariant (subsection 4.3)?

  4. 4.

How do we produce at least one experiment with which we can unambiguously show
that the wave function should not be extended in time (section 5)?

  5. 5.

Realistic experiments will need to run at ultra-short times and therefore high
energies. At these energies, particles may spring into existence from the
vacuum. How do we extend this analysis to the multiple particle case in a way
which is consistent with the work done in the single particle case (section
6)?

At attosecond times, we expect that the effects of dispersion in time will
dominate. Over picosecond and longer times we expect that the effects of the
clock time will need to be included.

Therefore, at attosecond times, we expect TQM will function as a kind of
lowest common denominator for the relativistic dynamics program. At picosecond
times, we may need to include contributions from the evolution parameter and
therefore be able to discriminate among various branches of the relativistic
dynamics program.

##  4 Free solutions

We now have the Schrödinger equation. What do its free solutions look like?

We examine in turn the birth, life, and death of a free particle. The
calculations are straightforward. But each stage will present problems
specific to TQM.

For convenience, we assemble the solutions of the free Schrödinger’s equation
in E.

###  4.1 Initial wave function

What do our wave functions look like at start?

We have a chicken and egg problem here. Any initial wave function had itself
to come from somewhere. How can we estimate the initial wave functions without
first knowing them?

We look specifically at the Klein-Gordon equation for a static electric
potential with no magnetic field:

${A_{0}}=\Phi\left({\vec{x}}\right),\vec{A}=0$ (98)

with potential:

$V\equiv q\Phi$ (99)

This includes attractive potentials, scattering potentials, and the free case
as the special case when $\Phi=0$.

In SQM the Klein-Gordon equation is:

$\left({{{\left({\imath{\partial_{\tau}}-V\left({\vec{x}}\right)}\right)}^{2}}-{\nabla^{2}}-{m^{2}}}\right){\bar{\psi}_{\tau}}\left({\vec{x}}\right)=0$
(100)

In TQM the equivalent is the four dimensional Schrödinger equation:

$\imath\frac{\partial}{{\partial\tau}}{\psi_{\tau}}\left({t,\vec{x}}\right)=-\frac{1}{{2m}}\left({{{\left({\imath{\partial_{t}}-V\left({\vec{x}}\right)}\right)}^{2}}-{\nabla^{2}}-{m^{2}}}\right){\psi_{\tau}}\left({t,\vec{x}}\right)$
(101)

The long, slow approximation picks out the solutions with:

$\left({{{\left({\imath{\partial_{t}}-V\left({\vec{x}}\right)}\right)}^{2}}-{\nabla^{2}}-{m^{2}}}\right){\psi_{\tau}}\left({t,\vec{x}}\right)=0$
(102)

We will assume we are already in possession of solutions to the SQM version of
the problem:

$\left(\left(E-V\left(\vec{x}\right)\right)^{2}-\vec{p}^{2}-m^{2}\right){\bar{\psi}_{E}}\left({\vec{x}}\right)=0$
(103)

This is necessarily in a specific frame, since SQM solutions are always (from
our perspective) taken with reference to a specific frame.

We leverage the SQM solutions in two different ways333This problem is also
solved in Fanchi [36], using different methods. to get the TQM solution:

  1. 1.

Separation of variables. Each SQM solution induces a TQM solution, which is
the SQM solution with a plane wave bolted on. This is technically correct, but
unphysical.

  2. 2.

Maximum entropy. We can use the long, slow approximation to estimate the mean
and uncertainty of the coordinate energy. With these we can use the method of
Lagrange multipliers to get the maximum entropy solution. Maximum entropy
solutions tend to be robust: even if we are wrong about the details, the order
of magnitude should be correct. This will provide our preferred starting
point.

As a quick check on the sanity of all this we will use the virial theorem to
estimate the TQM version of the atomic wave functions. The width in
time/energy of these matches the initial order of magnitude estimate we gave
in the introduction.

We will also show that while we have chosen a specific frame in which to
estimate the dispersion in time/energy we can get the estimate in an invariant
way.

####  4.1.1 Solution by separation of variables

We solve the TQM Klein-Gordon equation using separation of variables, looking
for solutions of the form:

$\varphi_{n}\left(t,\vec{x}\right)=\tilde{\phi}{}_{n}\left(t\right)\bar{\varphi}_{n}\left(\vec{x}\right)$
(104)

where $\bar{\varphi}_{n}\left(\vec{x}\right)$ is a solution of the SQM Klein-
Gordon equation:

$\left(\left(\bar{E}_{n}-V\left(\vec{x}\right)\right)^{2}-\vec{p}^{2}-m^{2}\right)\bar{\varphi}_{n}\left(\vec{x}\right)=0$
(105)

Then the coordinate time part is:

$\tilde{\phi}{}_{n}\left(t\right)=\frac{1}{\sqrt{2\pi}}e^{-\imath\bar{E}_{n}t}$
(106)

Because the potential is constant in time each use of the operator $E$ turns
into a constant $E_{n}$ via:

$E\to\imath\frac{\partial}{\partial t}\to\bar{E}_{n}$ (107)

We get immediately:

$\left(\left(\bar{E}_{n}-V\left(\vec{x}\right)\right)^{2}-\vec{p}^{2}-m^{2}\right)\varphi_{n}\left(t,\vec{x}\right)=0$
(108)

So every solution of the Klein-Gordon equation in SQM generates a
corresponding solution in TQM.

We could accomplish the same thing, formally, by taking $\tau\to t$. Since we
expect in general that $\left\langle t\right\rangle\approx\tau$ (discussed
further in the next section) this will often a give reasonable first
approximation.

However this is not entirely satisfactory. We have a solution which is “fuzzy”
in space, but “crisp” in time. A more realistic, if more complex solution,
would include off-shell components. Even if our wave function started out as a
simple plane wave in time, internal decoherence would rapidly turn it into
something a bit more cloud-like. While mathematically acceptable, our solution
is not physically plausible.

####  4.1.2 Solution by maximum entropy

The long, slow approximation picks out a single solution. But in practice we
expect there would be a great number of solutions, with the one given by the
long slow approximation merely the most typical.

Such a sum will have an associated probability density function. We can get a
reasonable first estimate of this by defining appropriate constraints and then
using the method of Lagrange multipliers to pick out the distribution with
maximum entropy.

From the probability density, we will infer the wave function.

##### Estimate of the probability density

We start with the free case, as representing the simplest case of a constant
potential. We treat the bound case below.

We assume we are given the SQM wave function in three dimensions. We use this
to compute the expectation of the energy, and the expectation of the energy
squared. These give us our constraints:

$\begin{array}[]{l}\left\langle 1\right\rangle=1\\\ \bar{E}\equiv\left\langle
E\right\rangle=\sqrt{m^{2}+\left\langle\vec{p}\right\rangle^{2}}\\\
\left\langle E^{2}\right\rangle=\left\langle
m^{2}+\vec{p}^{2}\right\rangle=m^{2}+\left\langle\vec{p}^{2}\right\rangle\end{array}$
(109)

The uncertainty in energy is defined as:

$\Delta E\equiv\sqrt{\left\langle{E^{2}}\right\rangle-{{\left\langle
E\right\rangle}^{2}}}$ (110)

The expectations are defined as integrals over the probability density:

$\left\langle f\right\rangle\equiv\int
d\vec{p}\bar{\rho}\left(\vec{p}\right)f\left(\vec{p}\right)$ (111)

The constraints imply $\Delta E=\Delta p$.

We will work with box normalized energy eigenfunctions:

$\phi_{n}\left(t\right)\equiv\frac{1}{\sqrt{2T}}e^{-\imath E_{0}nt}$ (112)
$E_{0}\equiv\frac{\pi}{T}$ (113)

Here $n$ is an integer running from negative infinity to positive and the
eigenfunctions are confined to a box extending $T$ seconds into the future and
$T$ seconds into the past, where $T$ is much larger than any time of interest
to us. (We will use a similar approach in developing the four dimensional Fock
space below, subsection 6.3.2).

A general wave function can be written as:

$\psi\left(t\right)=\sum\limits_{n=-\infty}^{\infty}c_{n}\phi_{n}\left(t\right)$
(114)

The coefficients $c$ only appear as the square:

$\rho_{n}\equiv c_{n}^{*}c_{n}$ (115)

Expressed in this language we have the constraints:

$\begin{array}[]{c}C_{0}\equiv\sum\limits_{n=-\infty}^{\infty}\rho_{n}-1=0\\\
C_{1}\equiv E_{0}\sum\limits_{n=-\infty}^{\infty}n\rho_{n}-\left\langle
E\right\rangle=0\\\ C_{2}\equiv
E_{0}^{2}\sum\limits_{n=-\infty}^{\infty}n^{2}\rho_{n}-\left\langle
E^{2}\right\rangle=0\end{array}$ (116)

We would like to find the solution that maximizes the entropy:

$S\equiv\sum\limits_{n=-\infty}^{\infty}-\rho_{n}\ln\left(\rho_{n}\right)$
(117)

We form the Lagrangian from the sum of the entropy and the constraints, with
Lagrange multipliers $\lambda_{0}$, $\lambda_{1}$, and $\lambda_{2}$:

$L\equiv S+\lambda_{0}C_{0}+\lambda_{1}C_{1}+\lambda_{2}C_{2}$ (118)

To locate the configuration of maximum entropy, we take the derivative with
respect to the $\rho_{n}$:

$\frac{\partial L}{\partial\rho_{n}}=0$ (119)

getting:

$-\ln\rho_{n}-1+\lambda_{0}+E_{0}n\lambda_{1}+E_{0}^{2}n^{2}\lambda_{2}=0$
(120)

Therefore the distribution of the $\rho_{n}$ is given by an exponential with
zeroth, first, and second powers of the energy:

$\rho\left(E\right)\sim e^{-a-bE-cE^{2}}$ (121)

The constraints force the constants:

$\rho\left(E\right)=\frac{1}{\sqrt{2\pi\Delta
E^{2}}}e^{-\frac{\left(E-\bar{E}\right)^{2}}{2\left(\Delta E\right)^{2}}}$
(122)

##### Estimate of the wave function

We therefore have the probability density in energy; now we wish to estimate
the corresponding wave function in energy (or equivalently time).

The simplest – and therefore best – way to do this is to write the total wave
function as the direct product of a wave function in energy times the
postulated wave function in momentum:

$\hat{\varphi}\left(E,p\right)\sim\hat{\tilde{\varphi}}\left(E\right)\hat{\bar{\varphi}}\left(p\right)$
(123)

We then use the Gaussian test function that matches the derived density
function for the energy factor:

$\hat{\tilde{\varphi}}_{0}\left(E\right)=\sqrt[4]{\frac{1}{\pi\sigma_{E}^{2}}}e^{\imath\left(E-E_{0}\right)t_{0}-\frac{\left(E-E_{0}\right)^{2}}{2\sigma_{E}^{2}}}$
(124)

with values:

$\sigma_{E}^{2}=\sigma_{p}^{2}$ (125)

and:

$E_{0}=\sqrt{m^{2}+\bar{p}^{2}}$ (126)

Taking the Fourier transform of the energy part we have:

$\tilde{\varphi}_{0}\left(t\right)=\sqrt[4]{\frac{1}{\pi\sigma_{t}^{2}}}e^{-\imath
E_{0}\left(t-t_{0}\right)-\frac{t^{2}}{2\sigma_{t}^{2}}}$ (127)
$\sigma_{t}^{2}=\frac{1}{\sigma_{E}^{2}}$ (128)

We set $t_{0}=0$ as the overall phase is already supplied by the
space/momentum part.

The full wave functions are the products of the coordinate time and space (or
energy and momentum) parts:

$\varphi_{0}\left(t,x\right)=\tilde{\varphi}_{0}\left(t\right)\bar{\varphi}_{0}\left(x\right)$
(129)
$\hat{\varphi}_{0}\left(E,p\right)=\hat{\tilde{\varphi}}_{0}\left(E\right)\hat{\bar{\varphi}}_{0}\left(p\right)$
(130)

####  4.1.3 Bound state wave functions

We extend this approach to estimate the dispersion of a bound wave function in
time.

In the case of a Coulomb potential we can estimate $\Delta p$ from the virial
theorem:

$\left\langle\frac{\vec{p}^{2}}{2m}\right\rangle=-\frac{1}{2}\left\langle
V\right\rangle$ (131)

which implies:

$\left\langle\frac{\vec{p}^{2}}{2m}\right\rangle+\left\langle
V\right\rangle=\bar{E}_{n}\to\left\langle\frac{\vec{p}^{2}}{2m}\right\rangle=-\bar{E}_{n}$
(132)

Since the average momentum is zero, we have the estimate of the uncertainty in
energy as:

$\Delta E_{n}=\sqrt{-2m\bar{E}_{n}}$ (133)

Substituting the mass of the electron and the Rydberg constant, we have for
the hydrogen ground state:

$\Delta E_{1}=\sqrt{2\cdot 13.6\textrm{eV}\cdot\left(0.511\cdot
10^{6}\right)\textrm{eV}}=3728\textrm{ eV}$ (134)

And the corresponding dispersion in coordinate time is:

$\Delta t=\frac{\hbar}{\Delta E_{n}}=.1766\textit{as}$ (135)

This matches the order of magnitude estimate we started with (subsection 1.2).
The numerical closeness is coincidental, but does increase confidence the
order of magnitude is correct.

We have therefore as the initial estimate of the TQM wave function for a
hydrogen atom:

$\begin{array}[]{c}{{\hat{\psi}}_{n}}\left({E,\vec{p}}\right)={{\hat{\varphi}}_{n}}\left(E\right){{\hat{\bar{\psi}}}_{n}}\left({\vec{p}}\right)\hfill\\\
{{\hat{\tilde{\varphi}}}_{n}}\left(E\right)=\sqrt[4]{{\frac{1}{{2\pi\hat{\sigma}_{n}^{2}}}}}{e^{\imath\left({E-{{\bar{E}}_{n}}}\right){t_{0}}-\frac{{{{\left({E-{{\bar{E}}_{n}}}\right)}^{2}}}}{{2\hat{\sigma}_{n}^{2}}}}\hfill}\\\
\hat{\sigma}_{n}^{2}=-4m{{\bar{E}}_{n}\hfill}\\\ \end{array}$ (136)

####  4.1.4 Frame independence of the estimate

We have given a reasonable estimate of the initial wave function in
time/energy. However, we have not yet established that the estimate is
independent of the initial choice of frame.

For both the bound and free cases, we choose our initial frame as the rest
frame. We make our estimate, via maximum entropy in this frame, then to get
the wave function in a arbitrary frame Lorentz transform to that frame:

$\psi^{\prime}\left({x^{\prime}}\right)=\psi\left({\Lambda x}\right)$ (137)

Since the rest frame is an invariant, the resulting initial wave functions are
well-defined.

###  4.2 Evolution of the free wave function

Now that we know what our starting wave functions look like, how do they
evolve over time?

For a first examination we work with the non-relativistic case; we will extend
to the relativistic case below (section 6). We will first look at the SQM
case, then the TQM case, then compare the two.

####  4.2.1 Evolution in SQM

We start with the familiar problem of the evolution of the non-relativistic
wave function with respect to clock time. We work with two dimensions –
$\tau,x$ – since the extension to $y,z$ is straightforward.

The non-relativistic Schrödinger equation is:

$\imath\frac{\partial}{{\partial\tau}}\bar{\psi}=\frac{{p^{2}}}{{2m}}\bar{\psi}=-\frac{{\partial_{x}^{2}}}{{2m}}\bar{\psi}$
(138)

At clock time zero we start with a Gaussian test function in momentum with
average position $x_{0}$, average momentum $p_{0}$, and dispersion in momentum
$\sigma_{p}$. To reduce clutter we use $p$ for $p_{x}$:

${{\hat{\bar{\varphi}}}_{0}}\left(p\right)=\sqrt[4]{{\frac{1}{{\pi\sigma_{p}^{2}}}}}{e^{-\imath
p{x_{0}}-\frac{{{\left({p-{p_{0}}}\right)}^{2}}}{{2\sigma_{p}^{2}}}}}$ (139)

In momentum space the problem is trivial. The solution is:

${{\hat{\bar{\varphi}}}_{\tau}}\left(p\right)=\sqrt[4]{{\frac{1}{{\pi\sigma_{p}^{2}}}}}{e^{-\imath
p{x_{0}}-\frac{{{\left({p-{p_{0}}}\right)}^{2}}}{{2\sigma_{p}^{2}}}-\imath\frac{{p^{2}}}{{2m}}\tau}}$
(140)

In coordinate space we get:

$\bar{\varphi}_{\tau}\left(x\right)=\sqrt[4]{\frac{1}{\pi\sigma_{x}^{2}}}\sqrt{\frac{1}{f_{\tau}^{\left(x\right)}}}e^{\imath
p_{0}x-\frac{1}{2\sigma_{x}^{2}f_{\tau}^{\left(x\right)}}\left(x-{x}_{0}-\frac{p_{0}}{m}\tau\right)^{2}-\imath\frac{p_{0}^{2}}{2m}\tau}$
(141)

with:

${\sigma_{x}}=\frac{1}{\sigma_{p}}$ (142)
$f_{\tau}^{\left(x\right)}=1+\imath\frac{\tau}{m\sigma_{x}^{2}}$ (143)

####  4.2.2 Evolution in TQM

In two dimensions the Schrödinger equation for TQM is:

$\imath\frac{\partial\psi_{\tau}}{\partial\tau}\left({t,\vec{x}}\right)=-\frac{{{E^{2}}-{{p}^{2}}-{m^{2}}}}{{2m}}{\psi_{\tau}}\left({t,\vec{x}}\right)=\left({\frac{{\partial_{t}^{2}}}{{2m}}-\frac{{\partial_{x}^{2}}}{{2m}}+\frac{m}{2}}\right){\psi_{\tau}}\left({t,\vec{x}}\right)$
(144)

We start in energy momentum space. The momentum part is as above. We start
with a Gaussian test function in energy, with average time at start $t_{0}$,
average energy $E_{0}$, and dispersion in energy $\sigma_{E}$:

${{\hat{\tilde{\varphi}}}_{0}}\left(E\right)\equiv\sqrt[4]{{\frac{1}{{\pi\sigma_{E}^{2}}}}}{e^{\imath
E{t_{0}}-\frac{{{\left({E-{E_{0}}}\right)}^{2}}}{{2\sigma_{E}^{2}}}}}$ (145)

Usually we will take $t_{0}=\tau_{0}=0$.

As a function of clock time we get:

${{\hat{\psi}}_{\tau}}\left({E,p}\right)={{\hat{\tilde{\varphi}}}_{0}}\left(E\right){{\hat{\bar{\varphi}}}_{0}}\left({p}\right)\exp\left({-\imath\frac{{{E^{2}}-{p^{2}}-{m^{2}}}}{{2m}}\tau}\right)$
(146)

We divide up the pieces of the clock time part, assigning the
$\frac{E^{2}}{2m}$ to the energy part, the $\frac{p^{2}}{2m}$ to the momentum
part, and keeping the third part outside:

${{\hat{\psi}}_{\tau}}\left({E,p}\right)={{\hat{\tilde{\varphi}}}_{\tau}}\left(E\right){{\hat{\bar{\varphi}}}_{\tau}}\left(p\right)\exp\left({\imath\frac{m}{2}\tau}\right)$
(147)

Now the energy part works in parallel to the momentum part:

${{\hat{\tilde{\varphi}}}_{\tau}}\left(E\right)\equiv\sqrt[4]{{\frac{1}{{\pi\sigma_{E}^{2}}}}}{e^{\imath
E{t_{0}}-\frac{{{\left({E-{E_{0}}}\right)}^{2}}}{{2\sigma_{E}^{2}}}-\imath\frac{{E_{0}^{2}}}{{2m}}\tau}}$
(148)

And in coordinate space:

$\tilde{\varphi}_{\tau}\left(t\right)=\sqrt[4]{\frac{1}{\pi\sigma_{t}^{2}}}\sqrt{\frac{1}{f_{\tau}^{\left(t\right)}}}e^{-\imath
E_{0}t+\imath\frac{E_{0}^{2}}{2m}\tau-\frac{1}{2\sigma_{t}^{2}f_{\tau}^{\left(t\right)}}\left(t-t_{0}-\frac{E_{0}}{m}\tau\right)^{2}}$
(149)

with the usual ancillary definitions:

${\sigma_{E}}=\frac{1}{{\sigma_{t}}}$ (150) $f_{\tau}^{\left(t\right)}\equiv
1-\imath\frac{\tau}{m\sigma_{t}^{2}}$ (151)

and with the expectation for coordinate time:

$\bar{t}_{\tau}={t}_{0}+\frac{E_{0}}{m}\tau$ (152)

implying a velocity for coordinate time with respect to laboratory time:

$\gamma=\frac{E_{0}}{m}$ (153)

In the non-relativistic case, $\gamma\approx 1$. So the expectation of the
coordinate time advances at the traditional one-second-per-second rate
relative to the clock time.

####  4.2.3 Comparison of TQM to SQM

The TQM and SQM approaches develop in close parallel. In both, the wave
functions are centered on the classical trajectory as given by the solution of
the Euler-Lagrange equations.

There is however one peculiarity.

We are relying on the long, slow approximation. Especially over short times,
this means that the total wave function appears to be relatively static with
respect to evolution in clock time:

$\imath\frac{\partial}{{\partial\tau}}\psi\approx 0$ (154)

In momentum space:

$f_{p}\equiv-\frac{{{E^{2}}-{p^{2}}-{m^{2}}}}{{2m}}\approx 0$ (155)

However, real wave functions are not static with respect to clock time.

The resolution is that most of the clock time dependence is carried by the
coordinate time:

$\frac{d}{{d\tau}}\psi=\frac{\partial}{{\partial\tau}}\psi+\frac{{dt}}{{d\tau}}\frac{\partial}{{\partial
t}}\psi\approx\frac{{dt}}{{d\tau}}\frac{\partial}{{\partial t}}\psi$ (156)

And in the non-relativistic case we have:

$\frac{{dt}}{{d\tau}}=\frac{E}{m}\approx 1$ (157)

So we get as a rough approximation:

$\frac{d}{{d\tau}}\psi\approx\frac{\partial}{{\partial t}}\psi$ (158)

We see that the expectation of the coordinate time is about equal to the clock
time. While Alice’s dog is always getting ahead of and behind her, on average
his position is about equal to hers:

$\left\langle t\right\rangle\approx\tau$ (159)

The Schrödinger equation gives the partial derivative with respect to clock
time; the total derivative behaves as expected.

###  4.3 Time of arrival measurements

Figure 3: Complex paths near the detector

So we know what our wave function looks like at start and how it evolves with
time. To complete the analysis of the free case we look at how it is detected.

We look specifically at the measurement of time-of-arrival. We assume we have
a particle going left to right, starting at $x=0$. We place a detector at
position $x=L$. It records when it detects the particle. The metric we are
primarily interested in is the dispersion in time-of-arrival at the detector.

In SQM, if a detector located at position $X$ registers a hit by a particle we
take the particle’s position in space as also $x=X$. Therefore in TQM, if
detector active at laboratory time $T$ registers a hit by a particle we must
take the particle’s position in time as $t=T$. This is required by our
principle of maximum symmetry between time and space.

By the same token, in SQM if an emitter located at position $X$ emits a
particle, we take the start position of the path as $x=X$. Therefore in TQM,
if an emitter active at laboratory time $T$ emits a particle, we must take the
start position in coordinate time as $t=T$.

In a practical treatment we would replace the phrases “at X” or “at T” with
“within the range $X\pm\frac{\Delta X}{2}$” and “within the range
$T\pm\frac{\Delta T}{2}$”.

If we know both source and detector positions in space and time then all
corresponding paths are clamped at both ends. In between source and detector
the paths can examine all sorts of interesting times and spaces but each path
is clamped at the endpoints. Alice and her dog leave from the same starting
point in space time and arrive at the same ending point in space time, but
while classical Alice takes the shortest path between the start and end
points, the quantum dog explores all paths.

##### Meaning of “all paths”

Paths in TQM are much more complex than those in SQM. If a detector is a
camera shutter, open for a fraction of a second, then any paths that arrive
early or late will merely be “eaten” by the closed shutter. But what if our
apparatus can somehow be toggled from transparent to absorptive and back, as
via “electromagnetically induced transparency” [67]? Then the paths can arrive
early but then circle back, or arrive late but circle forward, or even perform
a drunkard’s walk around the detector till they choose to fall into it.

Since we are primarily interested in comparisons of TQM to SQM, rather than in
fully exploring the elaborations of TQM, we will focus on the camera shutter
model. Paths that arrive before the shutter is open or after the shutter is
closed again will be silently absorbed by the camera itself. We defer to a
later investigation examination of more complex paths.

We employ the rest frame of the camera.

####  4.3.1 Metrics

With that dealt with, to compute the dispersion in time, we log how many hits
we get in each time interval (“clicks per tick”):

$\rho\left(\tau\right)$ (160)

then calculate the average:

$\left\langle\tau\right\rangle\equiv\int\limits_{-\infty}^{\infty}{d\tau\rho\left(\tau\right)}$
(161)

and the uncertainty:

${\left\langle{\Delta\tau}\right\rangle^{2}}\equiv\int\limits_{-\infty}^{\infty}{d\tau{\tau^{2}}\rho\left(\tau\right)}-{\left\langle\tau\right\rangle^{2}}$
(162)

####  4.3.2 Time of arrival in SQM

Figure 4: Time of arrival

We start with a particle with initial position $x=0$ and with average momentum
(in the $x$ direction) of $p_{0}$. We will assume that the initial dispersion
in momentum is small. We have the wave function from above. The probability
density is then:

$\bar{\rho}_{\tau}\left(x\right)\equiv\left|\bar{\varphi}_{\tau}\left(x\right)\right|^{2}=\sqrt{\frac{1}{\pi\sigma_{x}^{2}\left|f_{\tau}^{\left(x\right)}\right|^{2}}}e^{-\frac{\left(x-\frac{p_{0}}{m}\tau\right)^{2}}{\sigma_{x}^{2}\left|f_{\tau}^{\left(x\right)}\right|^{2}}}$
(163)

We assume there is a detector at position $x=L$. If the particle is released
at time $\tau=0$ the average time of arrival is:

$\bar{\tau}_{D}\equiv\frac{mL}{p_{0}}$ (164)

We can write the difference from the average time as
$\delta\tau\equiv\tau-\bar{\tau}$ We are interested in the uncertainty in time
of arrival at the detector or the expectation of ${\delta\tau}^{2}$.

If we were looking at measurements of the $x$ position we would know how to
proceed. We would compute:

$\Delta
x=\sqrt{\int{dx{{\left({x-\bar{x}}\right)}^{2}}}{\rho_{\tau}}\left(x\right)}$
(165)

We would like something not much more complex for the uncertainty in time. And
which makes sense for both SQM and TQM.

We will take an ad hoc approach here but then check it against the results of
a more detailed analysis by Muga and Leavens[68].

Normally we think of a wave function as something that evolves in time; it is
first a function of $\tau$, then of $x$. But here we are not interested in the
probability to be at $x$ at a specific time $\tau$; we are interested in the
probability to be at a specific clock time $\tau$ for a fixed $x=L$. So we
will rewrite the probability density as a function of $\tau$.

We will assume that we are dealing with a reasonably well-focused particle so
we may use a paraxial approximation. We can write $x=v\tau$ or $x=L+\delta x$.
We use this to rewrite the density function as a function of $\delta\tau$. We
take:

$L+\delta x=v\left({{{\bar{\tau}}_{D}}+\delta\tau}\right)\to\delta
x=v\delta\tau$ (166)

We keep only terms up through second order in ${\delta\tau}$.

We rewrite the numerator in the density function in terms of $\delta\tau$:

$\bar{\rho}_{\tau}\left(L\right)\approx\sqrt{\frac{1}{\pi\sigma_{x}^{2}\left|f_{\tau}^{\left(x\right)}\right|^{2}}}e^{-\frac{\left(v\delta\tau\right)^{2}}{\sigma_{x}^{2}\left|f_{\tau}^{\left(x\right)}\right|^{2}}}$
(167)

Since the numerator is already only of second order in $\delta\tau$ we need
only keep the zeroth order in $\delta\tau$ in the denominator:

$\sigma_{x}^{2}\left|f_{\tau}^{\left(x\right)}\right|^{2}=\sigma_{x}^{2}+\frac{\tau^{2}}{m^{2}\sigma_{x}^{2}}=\sigma_{x}^{2}+\frac{\left(\bar{\tau}+\delta\tau\right)^{2}}{m^{2}\sigma_{x}^{2}}\approx\frac{\left(\bar{\tau}+\delta\tau\right)^{2}}{m^{2}\sigma_{x}^{2}}\approx\frac{\bar{\tau}^{2}}{m^{2}\sigma_{x}^{2}}$
(168)

giving:

$\bar{\rho}_{\delta\tau}\approx\sqrt{\frac{v^{2}m^{2}\sigma_{x}^{2}}{\pi\bar{\tau}^{2}}}e^{-\frac{v^{2}m^{2}\sigma_{x}^{2}}{\bar{\tau}^{2}}\left(\delta\tau\right)^{2}}$
(169)

We define an effective dispersion in time:

$\bar{\sigma}_{\tau}\equiv\frac{1}{mv\sigma_{x}}\bar{\tau}$ (170)

And the probability of detection as:

$\bar{\rho}_{\delta\tau}=\sqrt{\frac{1}{\pi\bar{\sigma}_{\tau}^{2}}}e^{-\frac{\left(\delta\tau\right)^{2}}{\bar{\sigma}_{\tau}^{2}}}$
(171)

This is normalized to one, centered on $\tau=\bar{\tau}$, and with
uncertainty:

$\Delta\tau=\frac{1}{\sqrt{2}}\bar{\sigma}_{\tau}$ (172)

Particularly important is the inverse dependence on the velocity. Intuitively
if we have a slow moving (non-relativistic) particle, it will take a long time
to pass through by the $x$ position of the detector, causing the associated
uncertainty in time to be relatively large.

##### Comparison to a time-of-arrival operator

As a cross-check, we compare our treatment to the time-of-arrival operator
analysis in Muga and Leavens (who are following Kijowski [69]). They give a
probability density in time of:

$\rho\left(\tau\right)={\left|{\int\limits_{0}^{\infty}dp\sqrt{\frac{p}{m}}{e^{-\imath\frac{{{p^{2}}\tau}}{{2m}}}}\hat{\bar{\varphi}}\left(p\right)\exp\left({\imath
pL}\right)}\right|^{2}}+{\left|{\int\limits_{-\infty}^{0}dp\sqrt{\frac{{-p}}{m}}{e^{-\imath\frac{{{p^{2}}\tau}}{{2m}}}}\hat{\bar{\varphi}}\left(p\right)\exp\left({-\imath
pL}\right)}\right|^{2}}$ (173)

where $\hat{\bar{\varphi}}$ is an arbitrary momentum space wave function
normalized to one.

Assume:

${p_{0}}\gg{\sigma_{p}}$ (174)

so:

$p\approx{p_{0}}$ (175)

If our wave functions are closely centered on $p$, the term with negative
momentum can be dropped or even flipped in sign without effect on the value of
the integral. Further, by comparison to the exponential part, the term under
the square root is roughly constant:

$\sqrt{\frac{p}{m}}\approx\sqrt{\frac{p_{0}}{m}}$ (176)

So we may replace Muga and Leaven’s expression by the simpler:

$\rho\left(\tau\right)\approx\frac{{p_{0}}}{m}{\left|{\int\limits_{-\infty}^{\infty}dp{e^{-\imath\frac{{{p^{2}}\tau}}{{2m}}}}\hat{\bar{\varphi}}\left(p\right)\exp\left({\imath
pL}\right)}\right|^{2}}$ (177)

As the contents of the integral are the Fourier transform of the (clock) time
dependent momentum space wave function, it is the clock time dependent space
wave function at $x=L$:

$\rho\left(\tau\right)\approx\frac{{p_{0}}}{m}{\left|{{{\bar{\varphi}}_{\tau}}\left(L\right)}\right|^{2}}$
(178)

We are now in $x$ space. To get to $\tau$ space, we take:

$\int{\frac{{{p_{0}}}}{m}dx}=\int{vdx}=\int{d\tau}$ (179)

and the probability density from Muga and Leavens is now identical to ours.

####  4.3.3 Time of arrival in TQM

It is striking that there is considerable uncertainty in time even when time
is treated classically. Our hypothesized uncertainty in time will be added to
this pre-existing uncertainty.

Using the time wave function from equation 149 we have for the probability
density in time:

$\tilde{\rho}_{\tau}\left(t\right)=\sqrt{\frac{1}{\pi\sigma_{t}^{2}\left|f_{\tau}^{\left(t\right)}\right|^{2}}}e^{-\frac{1}{\sigma_{t}^{2}\left|f_{\tau}^{\left(t\right)}\right|^{2}}\left(t-\tau\right)^{2}}$
(180)

We multiply by the space part from above to get the full probability density:

$\rho_{D}\left(t,L\right)=\tilde{\rho}_{D}\left(t\right)\bar{\rho}_{D}\left(L\right)$
(181)

If $t$ were replaced by space dimension $y$, we would have no doubt as to how
to proceed. To get the overall uncertainty in $y$ we would integrate over
clock time:

$\left(\Delta y\right)^{2}=\int dy\left(y-\bar{y}\right)^{2}\int
d\tau\rho_{\tau}\left(y\right)\bar{\rho}_{\tau}\left(L\right)$ (182)

Therefore we write (taking $y\to t$):

$\left(\Delta t\right)^{2}=\int dt\left(t-\bar{\tau}\right)^{2}\int
d\tau\tilde{\rho}_{\tau}\left(t\right)\bar{\rho}_{\tau}\left(L\right)$ (183)

This is a convolution of clock time with coordinate time. To solve we first
invoke the same approximations as above:

$\begin{array}[]{c}\sigma_{t}^{2}\left|f_{\tau}^{\left(t\right)}\right|^{2}=\sigma_{t}^{2}+\frac{\tau^{2}}{m^{2}\sigma_{t}^{2}}=\sigma_{t}^{2}+\frac{\left(\bar{\tau}+\delta\tau\right)^{2}}{m^{2}\sigma_{t}^{2}}\approx\frac{\left(\bar{\tau}+\delta\tau\right)^{2}}{m^{2}\sigma_{t}^{2}}\approx\frac{\bar{\tau}^{2}}{m^{2}\sigma_{t}^{2}}\\\
\tilde{\sigma}_{\tau}\equiv\frac{\bar{\tau}}{m\sigma_{t}}\\\
\tilde{\rho}_{\tau}\left(t\right)\approx\sqrt{\frac{1}{\pi\tilde{\sigma}_{\tau}^{2}}}e^{-\frac{1}{\tilde{\sigma}_{\tau}^{2}}\left(t-\tau\right)^{2}}\end{array}$
(184)

So we have for the full probability distribution in $t$:

$\begin{array}[]{c}\rho_{\bar{\tau}}\left(t\right)\equiv\int
d\tau\sqrt{\frac{1}{\pi\tilde{\sigma}_{\tau}^{2}}}e^{-\frac{1}{\tilde{\sigma}_{\tau}^{2}}\left(t-\tau\right)^{2}}\sqrt{\frac{1}{\pi\bar{\sigma}_{\tau}^{2}}}e^{-\frac{1}{\bar{\sigma}_{\tau}^{2}}\left(\tau-\bar{\tau}\right)^{2}}\\\
\left(\Delta t\right)^{2}=\int
dt\left(t-\bar{\tau}\right)^{2}\rho_{\bar{\tau}}\left(t\right)\end{array}$
(185)

The convolution over $\tau$ is trivial giving:

${{\rho_{\bar{\tau}}}\left(t\right)=\sqrt{\frac{1}{{\pi\sigma_{\tau}^{2}}}}{e^{-\frac{{{\left({t-\bar{\tau}}\right)}^{2}}}{{\sigma_{\tau}^{2}}}}}}$
(186)

with the total dispersion in clock time being the sum of the dispersions in
coordinate time and in space:

${\sigma_{\tau}^{2}\equiv\tilde{\sigma}_{\tau}^{2}+\bar{\sigma}_{\tau}^{2}}$
(187)

This is intuitively reasonable.

The uncertainty is:

$\Delta\tau=\frac{1}{{\sqrt{2}}}\sqrt{\tilde{\sigma}_{\tau}^{2}+\bar{\sigma}_{\tau}^{2}}$
(188)

We collect the definitions for the two dispersions:

$\begin{array}[]{c}\bar{\sigma}_{\tau}^{2}=\frac{{{\bar{\tau}}^{2}}}{{{m^{2}}{v^{2}}\sigma_{x}^{2}}}\hfill\\\
\tilde{\sigma_{t}}^{2}\approx\frac{{{\bar{\tau}}^{2}}}{{{m^{2}}\sigma_{t}^{2}}}\hfill\end{array}$
(189)

From the long, slow approximation, we would expect particle wave functions to
have initial dispersions in energy/time comparable to their dispersions in
momentum/space. $\sigma_{t}\sim\sigma_{x}$. But the conventional contribution
has an additional $\frac{1}{v}$ in it. Since in the non-relativistic case,
$v\ll 1$ the total uncertainty will be dominated by the space part.

This helps to explain why dispersion in time has not been seen by accident. It
also motivates an exploration of the relativistic case, where the effects of
dispersion in time should be at least comparable to the effects of dispersion
in space (section 6).

##  5 Single and double slit experiments

There is already a significant literature on the “in time” versions of the
single and double slit experiments. The investigation of this problem started
over sixty years ago with Moshinsky [70, 71] and continues, with a recent
review by Gerhard and Paulus [72]. Particularly interesting for our purposes
are treatments of scattering of wave functions, as Umul [73] and Marchewka and
Schuss [74, 75].

Neither single nor double slit has, to the best of our knowledge, been solved
exactly. Approximations are necessary. Further the SQM and TQM branches have
to be approximated in a way that lets us compare the two branches directly. We
will treat the single slit first, then the double.

###  5.1 Single slit in time

The single slit in time experiment provides the decisive test of temporal
quantum mechanics.

In SQM, the narrower the slit, the less the dispersion in subsequent time-of-
arrival measurements. In TQM, the narrower the slit, the greater the
subsequent dispersion in subsequent time-of-arrival measurements. In
principle, the difference may be made arbitrarily great.

This distinction follows directly from the fundamental principles of quantum
mechanics. Picture a quantum wave function going through a gate in space. If
the gate is wide, diffraction by the edges is minimal and the subsequent
broadening of the wave function minimal. The gate will clip the beam around
the edges and that will be about it. But if the gate is narrow, then the wave
function will spread in a nearly circular pattern and the subsequent
broadening will be arbitrarily great.

In terms of the uncertainty principle, the gate represents a measurement of
the position. The narrower the gate, the less the uncertainty at the gate. If
$\Delta{y}$ is small, then $\Delta p_{y}$ must be correspondingly large and
the resulting spread greater at the detector. As $\Delta y\to
0\Rightarrow\Delta{p_{y}}\to\infty$. But a large $\Delta p_{y}$ implies – with
a bit of time – a large spread at the detector.

We translate this from space to time. We will start with a beam moving from
left to right in the $x$ direction, going through an extremely fast camera
shutter, and arriving at a detector. In both SQM and TQM, the faster the
shutter the smaller $\Delta t$. But in SQM the beam is clipped and the
dispersion at the detector correspondingly reduced. While in TQM the smaller
the $\Delta t$ the correspondingly greater the $\Delta E$. The greater the
$\Delta E$, the greater the dispersion in velocities and the greater the
dispersion in time-of-arrival at the detector.

The distribution of detections in clock time will give us the dispersion in
time-of-arrival, which is the key measurement.

We will ignore paths that loop back and forth through the gate. Most
elementary treatments make this assumption of a single passage (for an
analysis of the effects of multiple passages of a gate see Yabuki, Raedt, and
Sawant [76, 77, 78]).

We will also, following Feynman and Hibbs [64], take the gate as having a
Gaussian shape, rather then turning on and off instantly. This is more
realistic and avoids some distracting mathematical complexities that result
from using hard-edged gates.

Note also, in space we can make the gate entirely perpendicular to the beam.
Beam traveling in $x$, gate in $y$. The beam can start with zero momentum in
the $y$ direction, letting the $x$ momentum act as a carrier. But there is no
such thing as a particle that does not have at least some momentum in time,
i.e. energy. Therefore it is difficult to achieve a complete separation
between the measurement of $p_{x}$ and $E$. Here we will use increased
dispersion in the time-of-arrival measurements as the test of TQM. (Below we
will discuss briefly a way to separate the measurements of uncertainty in
space and in time: subsubsection 6.8.4.)

As usual we look first at SQM, then TQM.

####  5.1.1 Single slit in SQM

Figure 5: Single slit in SQM

We start by calculating the effects of a single slit in time on the dispersion
of time-of-arrival measurements in SQM.

The gate is located at $x=B$, centered on clock time $A$, with width in time
$W$:

$\overline{G}_{\tau}=e^{-\frac{\left(\tau-A\right)^{2}}{2W^{2}}}$ (190)

We start with a wave function in $x$:

$\bar{\varphi}_{\tau}\left(x\right)=\sqrt[4]{\frac{1}{\pi\sigma_{x}^{2}}}\sqrt{\frac{1}{f_{\tau}^{\left(x\right)}}}e^{\imath
p_{0}x-\frac{1}{2\sigma_{x}^{2}f_{\tau}^{\left(x\right)}}\left(x-x_{0}-\frac{p_{0}}{m}\tau\right)^{2}-\imath\frac{p_{0}^{2}}{2m}\tau}$
(191)

or in $p$:

${{\hat{\bar{\varphi}}}_{\tau}}\left(p\right)=\sqrt[4]{{\frac{1}{{\pi\sigma_{p}^{2}}}}}{e^{-\imath
p{x_{0}}-\frac{{{\left({p-{p_{0}}}\right)}^{2}}}{{2\sigma_{p}^{2}}}-\imath\frac{{p^{2}}}{{2m}}\tau}}$
(192)

We take the initial position $x_{0}=0$, initial velocity
$v\equiv\frac{p_{0}}{m}$. We assume the particle is non-relativistic so that
$v\ll 1.$

We assume that the incoming wave function can be treated as a sum of $p$ rays.
This worked in the time-of-arrival case, has the merit of simplicity, and lets
us make a direct comparison between TQM and SQM.

The evolution of the wave function is simplest in the $p$ basis, but the gate
is defined in the $\tau$ basis. Therefore we will need to shift back and forth
between the $p$ and the $\tau$ basis. To do this we use:

$p=m\frac{x}{\tau}$ (193)

with the ancillary definitions of $\delta p$ and $\delta\tau$:

$\begin{array}[]{c}p=p_{0}+\delta p=mv+\delta p\hfill\\\
\tau=\bar{\tau}+\delta\tau\hfill\end{array}$ (194)

For simplicity we center the particle beam on the gate. If $\bar{\tau}_{G}$ is
the average time at which the particle reaches the gate, we arrange it so
that:

$\overline{\tau}_{G}=A$ (195)

With the detector at position $x=L$, we define $T$ as the average time at
which the particle is detected. This gives:

$v=\frac{B}{A}=\frac{L}{T}$ (196)

We assume the beam is well-focused and use a paraxial approximation. To
quadratic order:

$\begin{array}[]{c}\delta
p=m\frac{x}{{\bar{\tau}+\delta\tau}}-m\frac{x}{{\bar{\tau}}}\approx
mv\left({-\frac{{\delta\tau}}{{\bar{\tau}}}+\frac{{\delta{\tau^{2}}}}{{{\bar{\tau}}^{2}}}}\right)\hfill\\\
\delta\tau=m\frac{x}{{p_{0}+\delta
p}}-m\frac{x}{p_{0}}\approx\frac{x}{v}\left({-\frac{{\delta
p}}{{p_{0}}}+\frac{{\delta{p^{2}}}}{{p_{0}^{2}}}}\right)\hfill\end{array}$
(197)

To first order:

$\frac{{\delta p}}{{{p_{0}}}}\approx-\frac{{\delta\tau}}{{\bar{\tau}}}$ (198)

$\delta p$ and $\delta\tau$ have opposite signs; faster means earlier, slower
means later.

If the gate is much wider than the wave function then the particle will pass
through unscathed. If the gate is narrower than the wave function then the
particle will clipped to the width of the gate.

The wave function just before the gate (using $p=\frac{{mB}}{{A+\delta\tau}}$)
is:

${{\bar{\varphi}}_{{G^{\left(pre\right)}}}}\left({\delta\tau}\right)=\sqrt[4]{{\frac{1}{{\pi\sigma_{G}^{2}}}}}{e^{-\frac{{\delta{\tau^{2}}}}{{2\sigma_{G}^{2}}}-\imath\frac{1}{{2m}}{{\left({\frac{{mB}}{{A+\delta\tau}}}\right)}^{2}}\left({A+\delta\tau}\right)}}$
(199)

with $\sigma_{G}$ defined by:

${\sigma_{G}}\equiv A\frac{{{\sigma_{p}}}}{{{p_{0}}}}$ (200)

Note that the wave function is correctly normalized for integration over
$\tau$. And that to lowest order:

$\frac{{\delta\tau}}{{{\sigma_{G}}}}=-\frac{{\delta p}}{{{\sigma_{p}}}}$ (201)

$\sigma_{G}$ is the measure of how wide the beam is in clock time when it
reaches the gate. It is linear in $A$ and proportional to $\sigma_{p}$.

For the SQM case, we assume no diffraction in time: the wave function will be
clipped by the gate, not diffracted. This means that to get the wave function
post-gate we must multiply the wave function pre-gate by the gate function:

${{\bar{\varphi}}_{{G^{\left(post\right)}}}}\left({\delta\tau}\right)={e^{-\frac{{{\left({\delta\tau}\right)}^{2}}}{2{W^{2}}}}}{{\bar{\varphi}}_{{G^{\left(pre\right)}}}}\left({\delta\tau}\right)$
(202)

Now we convert back to $p$ space, but on the far side of the gate:

${{\hat{\bar{\varphi}}}_{{G^{\left(post\right)}}}}\left(p\right)=\sqrt[4]{{\frac{1}{{\pi\sigma_{p}^{2}}}}}{e^{-\frac{{\delta{p^{2}}}}{{2\sigma_{p}^{2}}}-\frac{{A^{2}}}{{2{W^{2}}}}\frac{{\delta{p^{2}}}}{{p_{0}^{2}}}-\imath\frac{{p^{2}}}{{2m}}{\tau_{G}}}}$
(203)

The evolution from gate to detector only adds a phase:

${{\hat{\bar{\varphi}}}_{D}}\left(p\right)=\sqrt[4]{{\frac{1}{{\pi\sigma_{p}^{2}}}}}{e^{-\frac{{\delta{p^{2}}}}{{2\sigma_{p}^{2}}}-\frac{{\delta{p^{2}}}}{{2p_{0}^{2}}}\frac{{{A^{2}}}}{{{W^{2}}}}-\imath\frac{{{p^{2}}}}{{2m}}{\tau_{D}}}}$
(204)

We again switch to $\tau$ space at the detector. The significant change is
from ${\sigma_{p}}\to{{\sigma^{\prime}}_{p}}$ where the primed dispersion in
$p$ is:

$\frac{1}{{{{\left({{{\sigma^{\prime}}_{p}}}\right)}^{2}}}}=\frac{1}{{\sigma_{p}^{2}}}+\frac{{{A^{2}}}}{{{W^{2}}p_{0}^{2}}}$
(205)

With this we can write out the wave function at the detector by inspection. In
$p$ space:

${{\hat{\bar{\varphi}}}_{D}}\left(p\right)=\sqrt[4]{{\frac{1}{{\pi\sigma_{p}^{2}}}}}{e^{-\frac{{\delta{p^{2}}}}{{\left({{\sigma^{\prime}}_{p}}\right)}^{2}}-\imath\frac{{p^{2}}}{{2m}}{\tau_{D}}}}$
(206)

In $\tau$ space:

${{\bar{\varphi}}_{D}}\left({\delta\tau}\right)=\sqrt[4]{{\frac{1}{{\pi\sigma_{G}^{2}}}}}{e^{-\frac{{\delta{\tau^{2}}}}{{\left({{\sigma^{\prime}}_{G}}\right)}^{2}}-\imath\frac{{m{L^{2}}}}{{2T}}\left({1-\frac{{\delta\tau}}{T}+\frac{{\delta{\tau^{2}}}}{{T^{2}}}}\right)}}$
(207)

The primed dispersion in clock time at the detector is a scaled version of the
dispersion in momentum post gate:

${{\sigma^{\prime}}_{G}}\equiv T\frac{{{\sigma^{\prime}}_{p}}}{{p_{0}}}$ (208)

Note the factor $\sqrt[4]{{\frac{1}{{\pi\sigma_{G}^{2}}}}}$ is unchanged. The
ratio:

${\left({\frac{{{\sigma^{\prime}}_{G}}}{{\sigma_{G}}}}\right)^{2}}$ (209)

tells us what percentage of the particles get through the gate. But this does
not affect the calculation of the dispersion, which is normalized.

The effective dispersion of the wave function is proportional to $T$. It is
useful to scale that out. The result is controlled by the ratio of the angular
width of the beam in momentum space
${\theta_{p}}\equiv\frac{{{\sigma_{p}}}}{{{p_{0}}}}$ to the angular width of
the gate in time ${\theta_{G}}\equiv\frac{W}{A}$:

$\frac{{{{\left({{{\sigma^{\prime}}_{G}}}\right)}^{2}}}}{{{T^{2}}}}=\frac{{\theta_{p}^{2}\theta_{G}^{2}}}{{\theta_{p}^{2}+\theta_{G}^{2}}}$
(210)

When the gate is wide open, the dispersion at the detector is proportional to
the initial dispersion of the beam:

$W\to\infty\Rightarrow\frac{{{{\sigma^{\prime}}_{G}}}}{T}\to\frac{{{\sigma_{p}}}}{{{p_{0}}}}$
(211)

But when the gate is much narrower than the beam, the dispersion at the
detector proportional to the width of the gate:

$W\to 0\Rightarrow\frac{{{{\sigma^{\prime}}_{G}}}}{T}\to\frac{W}{A}$ (212)

The narrower the gate, the narrower the beam.

While the approximations we have used have been simple, this is a fundamental
implication of the SQM view of time, of time as classical. In SQM, wave
functions are not, by assumption, diffracted by a gate, they are clipped. And
therefore their dispersion in time must be reduced by the gate rather than
increased by it.

And we have therefore the uncertainty in time-of-arrival:

$\frac{{\Delta\tau}}{T}=\frac{1}{{\sqrt{2}}}\frac{{{{\sigma^{\prime}}_{G}}}}{T}$
(213)

####  5.1.2 Single slit in TQM

We now have a baseline from SQM; we turn to the single slit in time in TQM
using the SQM treatment as a starting point.

Figure 6: Single slit in TQM

By assumption, the gate will act only on the time part:

$\widetilde{G}_{t}=e^{-\frac{\left(t-A\right)^{2}}{2W^{2}}}$ (214)

For TQM, we start with a wave function factored in time and momentum:

${\psi_{\tau}}\left(t\right)={{\tilde{\varphi}}_{\tau}}\left(t\right){{\hat{\bar{\varphi}}}_{\tau}}\left(p\right)$
(215)

We are treating the $p$ part as a carrier, using the wave function from above.
The momentum part will be unchanged throughout.

We take the particle as starting with $t_{0}=0$. The wave function pre-gate
is:

${{\tilde{\varphi}}_{\tau}}\left(t\right)=\sqrt[4]{{\frac{1}{{\pi\sigma_{t}^{2}}}}}\sqrt{\frac{1}{{f_{\tau}^{\left(t\right)}}}}{e^{-\imath{E_{0}}t-\frac{1}{{2\sigma_{t}^{2}f_{\tau}^{\left(t\right)}}}{{\left({t-\frac{{{E_{0}}}}{m}\tau}\right)}^{2}}+\imath\frac{{E_{0}^{2}}}{{2m}}\tau}}$
(216)

We will again take the particle as non-relativistic: $E_{0}$ such that
$\frac{E_{0}}{m}\approx 1$.

On arrival at the gate the time part of the wave function is:

${{\tilde{\varphi}}_{{G^{\left(pre\right)}}}}\left({{t_{G}}}\right)=\sqrt[4]{{\frac{1}{{\pi\sigma_{t}^{2}}}}}\sqrt{\frac{1}{{f_{G}^{\left(t\right)}}}}\exp\left({-\imath{E_{0}}{t_{G}}-\frac{{{{\left({{t_{G}}-A}\right)}^{2}}}}{{2\sigma_{t}^{2}\left({1-\imath\frac{A}{{m\sigma_{t}^{2}}}}\right)}}+\imath\frac{{E_{0}^{2}}}{{2m}}A}\right)$
(217)

Post gate:

${{\tilde{\varphi}}_{{G^{\left(post\right)}}}}\left({t_{G}}\right)={e^{-\frac{{{\left({{t_{G}}-A}\right)}^{2}}}{{2W{{}^{2}}}}}}{{\tilde{\varphi}}_{{G^{\left(pre\right)}}}}\left({t_{G}}\right)$
(218)

The effect of the gate on the wave function is to rescale it:

$\frac{1}{2\sigma_{t}^{2}\left(1-\imath\frac{A}{m\sigma_{t}^{2}}\right)}\to\frac{1}{2W^{2}}+\frac{1}{2\sigma_{t}^{2}\left(1-\imath\frac{A}{m\sigma_{t}^{2}}\right)}$
(219)

We define rescaling constants $\sigma_{t}^{*2},{\tau^{*}}$ by:

$\frac{1}{{{W^{2}}}}+\frac{1}{{\sigma_{t}^{2}-\imath\frac{A}{m}}}=\frac{1}{{\sigma_{t}^{*2}-\imath\frac{{{\tau^{*}}}}{m}}}$
(220)

The effect of the gate is to change the shape of the wave function so that it
looks as if it had started at time $A-\tau^{*}$ with width $\sigma^{*}$. This
parameterization makes it easy to see what the gate is doing to the wave
function.

We have by inspection that as the width of the gate goes to infinity, it lets
the wave function through unchanged:

$W\to\infty\Rightarrow{{\sigma^{*}_{t}}}\to{\sigma_{t}},{\tau^{*}}\to A$ (221)

While as the gate gets narrow, the effective width of the wave function in
time goes to the width of the gate:

$W\to 0\Rightarrow{{\sigma^{*}_{t}}}\to W,{\tau^{*}}\to 0$ (222)

To compute the rescaling factors between these limits we clear the
denominators:

$\left({\sigma_{t}^{2}-\imath\frac{A}{m}}\right)\left({\sigma_{t}^{*2}-\imath\frac{{{\tau^{*}}}}{m}}\right)+{W^{2}}\left({\sigma_{t}^{*2}-\imath\frac{{{\tau^{*}}}}{m}}\right)={W^{2}}\left({\sigma_{t}^{2}-\imath\frac{A}{m}}\right)$
(223)

And equate the real and imaginary parts:

$\begin{array}[]{c}\left({\sigma_{t}^{2}+{W^{2}}}\right)\sigma_{t}^{*2}-\frac{A}{m}\frac{{{\tau^{*}}}}{m}={W^{2}}\sigma_{t}^{2}\hfill\\\
-\frac{A}{m}\sigma_{t}^{*2}-\left({\sigma_{t}^{2}+{W^{2}}}\right)\frac{{{\tau^{*}}}}{m}=-{W^{2}}\frac{A}{m}\hfill\\\
\end{array}$ (224)

This gives us a two by two matrix equation for $\sigma_{t}^{*2},{\tau^{*}}$
which we invert and apply to the right hand side:

$\left({\begin{array}[]{*{20}{c}}{\sigma_{t}^{*2}}\\\
{\frac{{{\tau^{*}}}}{m}}\end{array}}\right)=\frac{{{W^{2}}}}{D}\left({\begin{array}[]{*{20}{c}}{\sigma_{t}^{2}+{W^{2}}}&{-\frac{A}{m}}\\\
{-\frac{A}{m}}&{-\left({\sigma_{t}^{2}+{W^{2}}}\right)}\end{array}}\right)\left({\begin{array}[]{*{20}{c}}{\sigma_{t}^{2}}\\\
{-\frac{A}{m}}\end{array}}\right)$ (225)

with determinant:

$D={\left({\sigma_{t}^{2}+{W^{2}}}\right)^{2}}+\frac{{{A^{2}}}}{{{m^{2}}}}$
(226)

getting:

$\begin{array}[]{c}\sigma_{t}^{*2}=\frac{{{W^{2}}}}{D}\left({\left({\sigma_{t}^{2}+{W^{2}}}\right)\sigma_{t}^{2}+\frac{{{A^{2}}}}{{{m^{2}}}}}\right)\hfill\\\
{\tau^{*}}=\frac{A}{D}{W^{4}\hfill}\\\ \end{array}$ (227)

As a double check, we see we get the expected limits for $\sigma_{t}^{*2}$ and
${\tau^{*}}$ as $W\to\infty,W\to 0$.

$\sigma_{t}^{*2}$ and ${\tau^{*}}$ in turn gives us the $t$ wave function at
the detector. We replace $\sigma_{t}$ and $\tau_{G}$ in the initial wave
function by $\sigma^{*}_{t}$ and ${\tau^{*}}$:

${{\tilde{\varphi}}_{D}}\left({{t_{D}}}\right)=N\sqrt[4]{{\frac{1}{{\pi\sigma_{t}^{*2}}}}}\sqrt{\frac{1}{{{{f^{\prime}}_{G}}}}}{e^{-\imath{E_{0}}{t_{D}}-\frac{1}{{2\sigma_{t}^{*2}{{f^{\prime}}_{D}}}}{{\left({{t_{D}}-{{\bar{t}}_{D}}}\right)}^{2}}+\imath\frac{{E_{0}^{2}}}{{2m}}{\tau_{D}}}}$
(228)

with:

$\begin{array}[]{c}{{f^{\prime}}_{D}}\equiv
1-\imath\frac{{{\tau^{*}}+\left({T-A}\right)}}{{m\sigma_{t}^{*2}}}\hfill\\\
N\equiv\sqrt[4]{{\frac{{\sigma_{t}^{*2}}}{{\sigma_{t}^{2}}}}}\sqrt{\frac{{{{f^{\prime}}_{G}}}}{{{f_{G}}}}}\hfill\\\
\end{array}$ (229)

The value of $N$ drops out in the normalization when we calculation the
uncertainty.

The use of rescaling constants makes clear that the effect of the gate is to
reset the wave function, changing its shape and leaving no other memory of the
wave function before the gate. The wave function gets a fresh start with the
gate.

Of particular interest here is the limit $W\to 0$: an extremely narrow gate –
an extremely fast camera shutter – resets the wave function to have a width of
order width of the gate.

And therefore, given the Heisenberg uncertainty principle, the uncertainty in
energy post-gate goes as $\frac{1}{W}$. And this in turn forces the wave to
diverge more rapidly in time than would otherwise have been the case.

At large distances from the gate the dispersion of the wave function in time
$\tilde{\sigma}_{\tau}^{2}\equiv\sigma_{t}^{*2}{{f^{\prime}}_{\tau}}$ goes as:

${{\tilde{\sigma}}_{\tau}}\sim\frac{\tau}{m}{\sigma_{E}}=\frac{\tau}{{mW}}$
(230)

We can reapply the analysis from above (subsection 4.3.2) to get:

$\Delta\tau=\frac{1}{{\sqrt{2}}}\sqrt{\tilde{\sigma}_{\tau}^{2}+\bar{\sigma}_{\tau}^{2}}=\frac{1}{{\sqrt{2}}}\frac{T}{m}\sqrt{\frac{1}{{{W^{2}}}}+\frac{1}{{{v^{2}}\sigma_{x}^{2}}}}$
(231)

So in principle by making $W$ small enough we can make the uncertainty in time
of arrival as large as we wish.

Therefore:

  1. 1.

In SQM the uncertainty in clock time is directly proportional to the width of
the gate.

  2. 2.

In TQM the uncertainty in clock time is inversely proportional to the width of
the gate.

These effects are results of the fundamental assumptions. In SQM, time is a
parameter and the gate must clip the incoming wave function. In TQM, time is
an operator and the gate must diffract the incoming wave function.

Therefore the observable effect may be made, in principle, arbitrarily large.

To go from observable in principle to observable in practice will require us
to resolve two difficulties.

First we have to include the relativistic case: it is clear that the
$\frac{1}{v}$ factor will make it hard for the effects of uncertainty in time
to compete with the effects of uncertainty in space. This is a large part of
the motivation for looking at the multiple particle case in the next section:
relativistic velocities imply particle creation and therefore a need to
understand what TQM looks like when there is more than one particle in play.

Secondly we need a way to separate measurements of uncertainty in time from
measurements of uncertainty in space; an approach is suggested below
(subsubsection 6.8.4).

###  5.2 Double slit in time

The two principle effects we have been focused on are additional dispersion in
time and the uncertainty principle in time. We have not looked at effects
related to interference in time, mostly notably the double slit in time
experiment.

The double slit experiment has been described by Feynman [79] as the “only
mystery” of quantum mechanics. We therefore take a look at it in the TQM
context.

There have been many tests of the double slit experiment in space. The double
slit in time variation has been performed by Lindner et al [80]. In the Linder
et al experiment argon atoms are ionized using few-cycle electric pulses. If
the electron is ionized by a single peak, that scores as a single slit in
time; if by two, that scores as a double slit in time. The times are of order
500 attoseconds, much greater then the times we have been concerned with. But
the principles of the experiment are of great interest here.

Horwitz [38] argues that the analysis in Lindner et al is not properly
characterized as interference in time, since the analysis in Lindner et al is
based on SQM, in which time is not an operator.

But Horwitz then shows you get the same diffraction spacing using as a
starting point the relativistic dynamics equation444Note the sign of the
evolution parameter $s$ here is opposite from the sign used in Land and
Horwitz [37].:

$\imath\frac{\partial}{{\partial
s}}{\psi_{s}}=-\frac{{{E^{2}}-{{\vec{p}}^{2}}}}{{2m}}{\psi_{s}}$ (232)

As noted above, this differs from our Schrödinger equation by only a gauge
term. As gauge terms are not physically meaningful, Horwitz has provided us
with the diffraction pattern for TQM.

And as Horwitz gets the same spacing as Lindner did, the double slit in time
experiment does not help us differentiate between SQM and TQM, at least not to
first order555To higher orders, a preliminary examination suggests that while
the spacing peak-to-peak is unaffected, the widths of the individual peaks are
increased.. And therefore – somewhat surprisingly – the double slit in time
experiment does not help us to falsify TQM

And it follows in turn that it is not the double slit but the single slit
experiment which provides the decisive test of TQM.

##  6 Multiple particles

> “In what follows we assume that even though $A=\delta m$ is formally
> divergent, it is still `small´ in the sense that it is of the order of 1/137
> times the electron mass.” – J. J. Sakurai [81]

###  6.1 Why look at the multiple particle case?

In principle, we have enough to falsify TQM. Why then look at the multiple
particle case?

  1. 1.

From the above we see the effects of TQM will be largest at relativistic
velocities and short times. This implies we need to look at high energies and,
since high energies imply particle creation, at the multiple particle case.
(We thank Dr. Steve Libby for bringing this point to our attention.)

  2. 2.

There is the further concern that TQM may not be renormalizable. If the loop
integrals in SQM are barely renormalizable, with the addition of one more
dimension the loop integrals in TQM may well become completely intractable.

  3. 3.

Extending TQM to the multiple particle case opens up some new effects.

  4. 4.

And then there is the intrinsic interest of the question.

At the same time, there are formidable difficulties: the literature on quantum
field theory is vast and complex (among the references we have found helpful
are [82, 83, 84, 45, 85, 86, 87, 88, 89, 90, 91, 92, 52, 93, 94, 95, 40]).

We will focus on analyzing a toy model in a series of toy situations – but in
a way that will make clear how to extend the toy model to more realistic and
useful cases.

Since the complications of spin are inessential here, we will look at a simple
model with three massive spinless particles $A,$ $B$, and $C$ (correspondingly
very loosely to the electron, photon, and proton).

As in the single particle case, we will first work out the rules for the
various pieces of the path integral, then apply these rules to a few simple
cases. There is no possibility of covering all cases, but we will do enough to
make clear how to extend the approach to an arbitrary problem.

We will require, as before, manifest covariance and consistency with
established results. We will also require that the treatment of the single and
multiple cases be consistent; the single particle case should appear at the
end to be a specialized version of the multiple particle case.

The analysis divides into two parts:

First, we use a Lagrangian approach to derive the appropriate Feynman path
integrals. This approach will be loosely parallel to the approach in the
single particle case:

  1. 1.

Extend Fock space from 3D to 4D.

  2. 2.

Reuse the existing field theory Lagrangian densities by interpreting the time
as the coordinate time.

  3. 3.

Compute the action by integrating over both clock and coordinate times, so
that the integrals over the Lagrangian density goes from $\int{d\tau
d\vec{x}}\to\int{d\tau dtd\vec{x}}$.

  4. 4.

Verify that the free particle propagator computed with this approach matches
the single particle free particle propagator.

The result is that we get the TQM Feynman diagrams from their SQM equivalents
by keeping the topology of lines and vertexes the same, but replacing the SQM
parts with their TQM equivalents. We replace:

  1. 1.

SQM wave functions with TQM,

  2. 2.

SQM propagators with TQM,

  3. 3.

and SQM integrals over three space dimensions and clock time with TQM
integrals over the three space dimensions, coordinate time, and clock time.

In principle, we could simply have postulated these substitutions; the
derivation from first principles adds depth to our understanding, and also
makes possible the extension of TQM to problems not amenable to attack via the
Feynman diagram approach.

In the second part, we apply the derived Feynman rules to a few simple cases:

  1. 1.

Free particle,

  2. 2.

Emission of a particle,

  3. 3.

Absorption of a particle,

  4. 4.

Exchange of a particle,

  5. 5.

Loop correction to the mass.

By combining these elements we can in principle calculate any Feynman diagram.

The loop calculation provides a preliminary success for TQM. If treated
naively, the loop diagrams in TQM are divergent. But the combination of Morlet
wavelet decomposition and entanglement in time – neither alone sufficient –
causes the loop integrals to converge.

As we develop the multiple particle case, we will see a number of additional
effects:

  1. 1.

anti-symmetry in time,

  2. 2.

forces of anticipation and regret,

  3. 3.

interference and entanglement in time.

With the approach taken here we have no free or adjustable parameters. We have
a simple transition from the single to the multiple particle treatment. We do
not have the familiar ultraviolet divergences. And we have a large number of
opportunities for experimental test.

###  6.2 $ABC$ model

Figure 7: Interactions in the $ABC$ model

The simplest model we can find that lets us cover the basic interactions has
three spinless, massive particle species $A,$ $B$, and $C$. We assign them
non-zero masses $m$, $\mu$, and $M$ respectively. We will take $\mu$ as small
as it needs to be. They are real fields. $A$’s emit and absorb $B$’s with
amplitude $\lambda$. $C$’s emit and absorb $B$’s with amplitude $\Lambda$.
$A$’s and $C$’s do not talk with each other directly. There are no other
interactions.

Loosely, $A$ is a spinless model of an electron, $B$ of a photon, and $C$ of a
proton.

We will focus primarily on the $A$ and $B$ particles. We will need the $C$ for
the discussion of particle exchange.

###  6.3 Fock space

In the single particle case we generalized the wave functions from three
dimensions to four. Here we generalize Fock space from three dimensions to
four. As paths may be seen as a series of wave functions, one wave function
per clock tick, this implicitly generalizes the associated paths from three
dimensions to four as well.

We use box normalization. The box runs from $-L\to L$ in all four coordinates:
coordinate time and the three space dimensions. It is taken to extend well
past the wavelets we are working with.

Using box normalization means that the Fourier transforms will be discrete.
Discrete Fourier transforms are convenient for discussing various points of
principle and to help in visualizing the field theory calculations. For the
actual calculations we will use continuous wave functions.

In general the extrusion from three to four dimensions is straight forward; a
few specific points require attention.

####  6.3.1 Fock space in three dimensions

Figure 8: Fock space in three dimensions

##### Single particle basis wave functions

We will focus on the $x$ coordinate here; $y$ and $z$ are the same. We break
our box into 2M pieces, implying a lattice spacing $a\equiv\frac{L}{M}$. $a$
has dimensions of length. This lets us replace the smoothly varying $x$ with
the values of $x$ at a series of points:

$x\in\left[{-L,L}\right]\to x=\left({-aM,\ldots,0\ldots,aM}\right)$ (233)

The continuous/discrete translation table is:

$\left({x,y,z}\right)\leftrightarrow\left({ai,aj,ak}\right)$ (234)

Integrals over space go to sums over $i,j,k$:

$\int\limits_{-L}^{L}{dxdydz}\leftrightarrow{a^{3}}\sum\limits_{i,j,k=-M}^{M}{}$
(235)

The coordinate basis is trivial, just Kronecker $\delta$ functions:

${\phi_{\vec{x}^{\prime}}}\left({\vec{x}}\right)\equiv{\delta^{3}}\left({\vec{x}-\vec{x}^{\prime}}\right)\leftrightarrow{\delta_{ii^{\prime}}}{\delta_{jj^{\prime}}}{\delta_{kk^{\prime}}}$
(236)

We assume the wave functions are periodic in $2L$. The periodic condition is
not important; $L$ will be chosen large enough that all interesting wave
functions are well inside of it. They will be trivially periodic because they
are zero on both sides of each dimension.

We normalize the basis wave functions to one:

$\int\limits_{-L,-L,-L}^{L,L,L}{d\vec{x}\phi_{\vec{k}}^{*}\left({\vec{x}}\right){\phi_{\vec{k}^{\prime}}}\left({\vec{x}}\right)}={\delta_{\vec{k}\vec{k}^{\prime}}}$
(237)

giving:

${\phi_{\vec{k}}}\left({\vec{x}}\right)=\frac{1}{{{\sqrt{2L}}^{3}}}\exp\left({\imath\vec{k}\cdot\vec{x}}\right)$
(238)

Now we can expand an arbitrary wave function in terms of the basis functions:

$\phi\left({\vec{x}}\right)=\sum\limits_{\vec{k}}{c_{\vec{k}}}{\phi_{\vec{k}}}\left({\vec{x}}\right)$
(239)

The measure in the path integrals is in terms of the $c$’s:

$\mathcal{D}\phi\equiv\prod\limits_{n=0}^{N}{{\mathcal{D}_{n}}\phi},{\mathcal{D}_{n}}\phi\equiv\prod\limits_{\vec{k}}{d{c_{\vec{k}}}}$
(240)

so there is one set of space integrals at each clock tick.

At the end of the discrete part of the calculation we will be letting $M$,
$N$, and $L$ go to infinity. As the effects of TQM are averaged out over
larger times we will not be letting $T$ go to infinity, even in the SQM case.

In the continuum limit we have:

${\phi_{\vec{k}}}\left({\vec{x}}\right)\to\frac{1}{{{\sqrt{2\pi}}^{3}}}\exp\left({\imath\vec{k}\cdot\vec{x}}\right)$
(241)

##### Multi-particle wave functions

Things become interesting when we go to multiple particle wave functions. For
two particles:

${\phi_{\vec{k}\vec{k}^{\prime}}}\left({1,2}\right)\equiv\frac{1}{{\sqrt{2}}}\left({{\phi_{\vec{k}}}\left(1\right){\phi_{\vec{k}^{\prime}}}\left(2\right)+{\phi_{\vec{k}}}\left(2\right){\phi_{\vec{k}^{\prime}}}\left(1\right)}\right)$
(242)

As we get to larger and larger numbers of particles, these wave functions
become tricky to write out and manage. To simplify, we use the familiar
annihilation and creation operators, defined by:

$\begin{array}[]{c}a_{\vec{k}}^{{\dagger}}\left|{n_{\vec{k}}}\right\rangle=\sqrt{{n_{\vec{k}}}+1}\left|{{n_{\vec{k}}}+1}\right\rangle\hfill\\\
{a_{\vec{k}}}\left|{n_{\vec{k}}}\right\rangle=\sqrt{{n_{\vec{k}}}}\left|{{n_{\vec{k}}}-1}\right\rangle\hfill\end{array}$
(243)

with the usual commutation operators:

$\left[{{a_{\vec{k}}},a_{\vec{k}^{\prime}}^{{\dagger}}}\right]={\delta_{\vec{k}\vec{k}^{\prime}}}$
(244)

We define the single particle operator:

$\bar{\phi}\left({\vec{x}}\right)=\sum\limits_{\vec{k}}{{a_{\vec{k}}}\bar{\phi}_{\vec{k}}^{{\dagger}}\left({\vec{x}}\right)+a_{\vec{k}}^{{\dagger}}{{\bar{\phi}}_{\vec{k}}}\left({\vec{x}}\right)}$
(245)

An arbitrary multiple particle basis state may be built up as products of
these:

$\left|{\left\\{{n_{\vec{k}}}\right\\}}\right\rangle=\frac{1}{{\sqrt{\prod\limits_{\vec{k}}{{n_{\vec{k}}}!}}}}\prod\limits_{\vec{k}}{{\left({a_{\vec{k}}^{{\dagger}}}\right)}^{{n_{\vec{k}}}}}\left|0\right\rangle$
(246)

where ${\left\\{{n_{\vec{k}}}\right\\}}$ is a specific set of occupation
numbers.

We define the general wave function in the occupation number basis:

$\sum\limits_{\left\\{{n_{\vec{k}}}\right\\}}{c_{\left\\{{n_{\vec{k}}}\right\\}}}\left|{\left\\{{n_{\vec{k}}}\right\\}}\right\rangle$
(247)

with normalization:

$1=\sum\limits_{\left\\{{{n_{\vec{k}}}}\right\\}}{c_{\left\\{{{n_{\vec{k}}}}\right\\}}^{2}}$
(248)

Note we are not defining the creation and annihilation operators in terms of
an infinite set of harmonic oscillators; we are defining them by their effects
on Fock space. We can think of movement in Fock space as a giant game of
snakes and ladders, with the creation operators as the ladders, the
annihilation operators as the snakes.

####  6.3.2 Fock space in four dimensions

Figure 9: Fock space in four dimensions

##### Single particle basis wave functions

The development in TQM works along the same lines but with one more dimension
$t$ and a fourth index $h$. We are treating coordinate time essentially like a
fourth spatial dimension, much like the $x_{4}\equiv\imath ct$ trick of
earlier works on special relativity. Paths in TQM will typically start at
$\tau=0$ and finish at some defined clock time $\tau=T$. But their paths in
coordinate time may well dive before $t=0$ and climb past $t=T$. To make sure
that all of the relevant paths are included in our box, we require that $-L\ll
0$ and $L\gg{T}$. Using the same $L$ for time as for space, and requiring that
$L\to\infty$ accomplishes this.

The continuous/discrete translation table is now: