# The Anomalous Nambu-Goldstone Theorem in Relativistic/Nonrelativistic
Quantum Field Theory

Tadafumi Ohsaku

(today)

In der Welt habt ihr Angst; aber seid getrost, ich habe die Welt überwunden. (
Johannes, Kapitel 16 )

It always seems impossible until it is done. ( Nelson Mandela )

This Paper is Dedicated for Our Brave Fighters and Super-Heroes for the
Fundamental Human Rights Around the World.

Abstract: The anomalous Nambu-Goldstone ( NG ) theorem which is found as a
violation of counting law of the number of NG bosons of the standard ( normal
) NG theorem in nonrelativistic and Lorentz-symmetry-violated relativistic
theories is studied in detail, with emphasis on its mathematical aspect from
Lie algebras, geometry to number theory. The basis of counting law of NG
bosons in the anomalous NG theorem is examined by Lie algebras ( local ) and
Lie groups ( global ). A quasi-Heisenberg algebra is found generically in
various symmetry breaking schema of the anomalous NG theorem, and it indicates
that it causes a violation/modification of the Heisenberg uncertainty relation
in an NG sector which can be experimentally confirmed. This fact implies that
we might need a framework ”beyond” quantum mechanical apparatus to describe
quantum fluctuations in the phenomena of the anomalous NG theorem which might
affect formations of orderings in quantum critical phenomena. The formalism of
effective potential is presented for understanding the mechanism of anomalous
NG theorem with the aid of our result of Lie algebras. After an investigation
on a bosonic kaon condensation model with a finite chemical potential as an
explicit Lorentz-symmetry-breaking parameter, a model Lagrangian approach on
the anomalous NG theorem is given for our general discussion. Not only the
condition of the counting law of true NG bosons, but also the mechanism to
generate a mass of massive NG boson is also found by our examination on the
kaon condensation model. Furthermore, the generation of a massive mode in the
NG sector is understood by the quantum uncertainty relation of the Heisenberg
algebra, obtained from a symmetry breaking of a Lie algebra, which realizes in
the effective potential of the kaon condensation model. Hence the relation
between a symmetry breaking scheme, a Heisenberg algebra, a mode-mode
coupling, and the mechanism of mass generation in an NG sector is established.
Finally, some relations between the Riemann hypothesis and the anomalous NG
theorem are presented.

KEYWORDS: Normal and anomalous Nambu-Goldstone theorem, Lie algebras and Lie
groups, Heisenberg algebra, the Heisenberg uncertainty relation, quantum
fluctuation and phase transition, differential geometry, number theory, the
Riemann hypothesis, spin systems, QCD.

##  1 Introduction

This paper belongs to the recent efforts to intend to make a final answer on
the controversial ( or, not well-organized ) issue on the counting law of the
number of Nambu-Goldstone ( NG ) bosons/modes in a nonrelativistic and/or
Lorentz-symmetry-broken system. We mainly concentrate on the cases of so-
called continuous internal symmetries expressed by Lie groups, commute with
the spacetime 4-translations and the Lorentz 4-rotations. We restrict
ourselves on the cases of four-dimensional spacetime: Two- and three-
dimensional cases are outside of our discussion due to the Mermin-Wagner-
Coleman theorem [11,48,52]. The Nambu-Goldstone theorem ( established around
1960-1962 [26,27,60,61] ) states that a zero-mass/gap particle ( Nambu-
Goldstone boson ) naturally arises from a theory associated with a
spontaneously-broken-symmetry generator, with one-to-one correspondences
between broken generators and the NG bosons. Here we call this standard
situation as the ”normal” NG theorem. It is a famous fact that, in an
anomalous case, the number of NG bosons does not coincide with the number of
broken generators of ${\rm Lie}(G)$ in a case of nonrelativistic field theory
( $G$ is a Lie group which gives a symmetry of the system, Lie$(G)$ denotes
its corresponding Lie algebra ). Nielsen and Chadha found this phenomena in
1976 [64]. After those discoveries in fundamental physics, almost four
decades, there is no precise solution or any explanation of reasoning of the
Nielsen-Chadha anomaly. Quite recently ( 2002-2013 ), this controversy has
been started to be resolved by several works in theoretical physics.

In this section, we summarize those recent results of their works appeared in
literature. To clarify our discussions and perspectives, one should employ a
classification: Lorentz = symmetric, spontaneously broken, explicitly broken;
Lie group = symmetric, spontaneously broken, explicitly broken. Thus totally
$3\times 3=9$ cases we have in general. In those cases, we consider here some
examples where the Lorentz symmetry is explicitly broken, while a Lie group
symmetry is spontaneously/explicitly broken. We will show that these two
examples are understood by a single framework. In principle, our anomalous NG
theorem is fall into the category of ”spontaneous” cases without an explicit
symmetry breaking. Other cases remain for our further study in future. In
fact, recent several works given in literature also consider the cases that
the Lorentz symmetry is explicitly broken or a nonrelativistic case, while a
Lie group of internal symmetry is spontaneously broken.

First, we discuss the main result of Refs.[8,34,81,82,83,84] which is
summarized into the following inequality for the number of NG bosons given by
Watanabe and Brauner: They gave it without any derivation [81]:

$\displaystyle n_{NG}$ $\displaystyle\geq$ $\displaystyle
n_{BS}-\frac{1}{2}{\rm rank}\langle 0|[Q^{A},Q^{B}]|0\rangle.$ (1)

Here, $n_{NG}$ gives the number of NG bosons they would be observed ( when a
norm is positive and physical ), $n_{BS}$ is the number of broken generators
of ${\rm Lie}(G)\simeq T_{e}G$ ( $G$; a Lie group which describes a symmetry
of system we consider, $e$; the origin ), $Q^{A}$ and $Q^{B}$ (
$A,B=1,\cdots,n_{BS}$ ) imply the conserved charges of broken generators, and
$|0\rangle$ is the vacuum of a theory. It should be mentioned that the rank is
defined for a matrix with matrix entries of indices $AB$. While, we know the
very simple and universal law that the number of broken generators $n_{BS}$
equals the sum of the number of ”true” NG bosons and the number of ”massive”
NG bosons:

$\displaystyle n_{true-NG}$ $\displaystyle=$ $\displaystyle n_{BS}-n_{massive-
NG}.$ (2)

It is clear for us that $\frac{1}{2}{\rm rank}\langle[Q^{A},Q^{B}]\rangle$
must give the number of massive modes ( the rank of Hessian of effective
action expanded by NG modes around a VEV, see the paper of S. Weinberg in our
references [86] ): The matrix is given in a quadratic form of broken
generators. Thus, our issue is now understood how to find a law in which the
broken generators contain massive modes. Let us compare the anomalous behavior
with a case of explicit symmetry breaking. The paper of Weinberg [86]
discusses a chiral Lagrangian with an explicit symmetry breaking mass term
proportional to

$\displaystyle{\rm tr}(e^{iQ}\Phi^{-iQ})$ $\displaystyle=$ $\displaystyle{\rm
tr}(g\Phi g^{-1})={\rm tr}{\rm Ad}(G(\Phi)),$ (3)

( $\Phi$; some bosonic fields ) and gives a formula of square of mass
parameters of NG bosons by the second-order derivative of the mass term. It is
given by the bracket like the following form, namely, a term given by twice
actions of adjoints:

$\displaystyle[[\Phi,Q^{A}],Q^{B}]$ (4)

and if we regard this as a matrix of indices $(A,B)$, then its dimension is
exactly equal with $[Q^{A},Q^{B}]$ appeared in the formula for counting the
number of NG bosons. Therefore, there is a similarity between an anomalous and
an explicit symmetry breaking at the Lie algebra level beside the factor 1/2.
Later, we will see how this factor 1/2 arises in our anomalous NG theorem. If
a theory has $n_{ex}$ explicit symmetry breaking mass parameters, and if

$\displaystyle n_{ex}\geq\frac{1}{2}{\rm rank}\langle[Q^{A},Q^{B}]\rangle,$
(5)

then

$\displaystyle n_{NG}=n_{BS}-n_{ex}$ (6)

may holds, since an explicit symmetry breaking parameter enforces that an NG
boson always has a finite mass.

While, after the work of Watanabe and Brauner, Hidaka [34] derived the
equation ( replaces $\geq$ to $=$ in the above inequality (1) ) via the
generalized Langevin formalism: His formalism is essentially the same with the
method of effective action. Thus, later we utilize the effective action
formalism, both Lorentz-violating relativistic and nonrelativistic cases.

Note that in a Poincaré invariant theory, a charge $Q$ is Lorentz scalar when
it is conserved, $[P^{\mu},Q]=0$. This is due to the Coleman-Mandula no-go
theorem [12]. Schaefer et al. pointed out in their paper [77] that if $Q$ is
given from the zeroth-component of conserved vector current of an internal
symmetry, then it cannot have a nonvanishing VEV for a Lorentz-symmetric
vacuum. They argue that we need $\langle Q\rangle\neq 0$ for realizing an
anomalous behavior of NG theorem. We need a careful discussion on it. In fact,
we now consider a theory of Lorentz-violating system, hence we cannot restrict
$Q$ as a Lorentz scalar. Moreover, we should distinguish the cases of Wigner
phase and Nambu-Goldstone ( NG ) phase. In a Wigner phase, $Q|0\rangle=0$ is
immediately concluded since $Q$ is a conserved quantity, while $Q|0\rangle\neq
0$ in an NG phase, and it does not look like a conserved quantity: This is the
essential part of the NG theorem. The symmetry of a Lagrangian ( theory ) and
its vacuum do not coincide with each other in an NG phase. Due to the unitary
inequivalence, one can not definitely say about what $e^{iQ}|0\rangle$ gives.
For example, the vector $Q|0\rangle$ cannot be normalized in an NG phase. It
might be possible to say that $|0\rangle$ or $Q|0\rangle$ are not $G$-modules
in the naive sense. While, if a theory spontaneously breaks its vacuum
symmetry, and if the Lorentz symmetry is broken under a certain manner, then
we lost the basis of the statement of $\langle 0|Q|0\rangle=0$ even though it
will be defined as an integral of three-dimensional total volume/space: The
physical situation of those VEVs may be determined self-consistently. We
emphasize the fact that this discussion is valid for a quantum theory but not
for classical systems, since we take a VEV of a quantum operator. We will also
give some insights on the case where a broken charge is a Lorentz-violating
generic tensor. Hence we have another classification: Vacuum = Lorentz
symmetric/Lorentz violated, Nöther charge = Lorentz symmetric/Lorentz
violated.

According to our short observation of several previous results in literature,
we classify NG bosons into, (i) true NG bosons as exactly massless particles,
(ii) massive NG bosons in our anomalous NG theorem, (iii) pseudo NG bosons
which have finite masses due to explicit symmetry breaking parameters in the
Lagrangian of the beginning. Several examples of (iii) have been studied, for
example, in Refs. [16,70]. Our terminology presented here is not the same with
the famous classification of NG bosons given by S. Weinberg for Lorentz-
invariant relativistic cases [86]: Our present discussion should be understood
as a generalization/extension of it.

Our several classifications are summarized into the table given in the next
page:

Theory (Lagrangian) | LS, explicitly LV
---|---
Vacuum | LS, spontaneously LV, explicitly LV
Lie group | symmetric, SB, AB, EB
Discrete ( C, P, T ) | symmetric, SB, EB
Nöther charge | LS, LV
NG boson | true, anomalously massive, pseudo

Here, several abbreviations mean: LS = Lorentz symmetric, LV = Lorentz
violated, SB = spontaneously broken, AB = anomalously broken, EB = explicitly
broken. ”Discrete” indicates a discrete symmetry, typically as a charge
conjugation ( C ), a parity ( P ), and a time-reversal ( T ).

This paper is organized as follows: In sec. 2, several typical symmetry
breaking schema will be studied from their Lie algebra/group aspects, and will
find several characteristic features of them in our anomalous NG theorem,
which never takes place in the standard NG case. In sec. 3, an effective
potential formalism will be investigate to understand the mechanism of our
anomalous NG theorem by employing our Lie-algebra results. In sec. 4, a kaon
condensation model with a finite chemical potential will be examined to obtain
our viewpoint on a generic Lagrangian of NG bosons which gives the phenomenon
of anomalous NG theorem. Then we will construct a generic Lagrangian which
cause the anomalous NG theorem, in sec. 5. Some relations between our
anomalous NG theorem and number theory, especially the Riemann zeta function,
will be presented in sec. 6. Finally, the concluding remarks will be given in
sec. 7.

##  2 Lie Algebras, Lie Groups, and Symmetry Breaking Schema

In this section, our anomalous NG theorem is examined by Lie algebras ( give
some local characters of NG sectors ) and Lie groups ( contain informations on
global aspects/structures of NG manifolds ), with employing several examples.

First, we would like to pay attention on the following fact before obtaining a
general discussion of symmetry breakings. In a breaking scheme of a symmetry,
it is not always the case that a Lie group $G$ is broken to a Lie subgroup $H$
to give a coset $G/H$: Thus, an examination on cosets as results of symmetry
breakings is not enough for studying the ( global ) nature of (
normal/explicit+dynamical/anomalous ) NG theorem. For example, let us consider
some examples of $SO(3)$ or $SU(2)$. ( You can compare with the case of
$U(2)$, or the electroweak symmetry breaking of the Standard Model! ) The
Hamiltonian of spin systems of ferro- and antiferromagnets may belong to
$SO(3)$ and sometimes $SU(2)$ [87] ( they are locally isomorphic, ${\rm
Lie}SO(3)\simeq{\rm Lie}SU(2)\simeq{\rm Lie}USP(2)$, and thus one has a
freedom to choose one of them at least at the Lie algebra level ). The isospin
space also be described by $SU(2)$. In a ferromagnetic case, quite a lot of
works consider broken generators as $s_{1}$ and $s_{2}$ of $SU(2)$ while
$s_{3}$ remains ”unbroken.” ( $s_{a}$, $a=1,2,3$ $\in{\rm Lie}(SU(2))$, by
using the representation of Pauli matrices. ) This breaking scheme is
schematically denoted as $SU(2)\to U(1)$, but it does not give a coset: This
breaking scheme does not have a coset ( quotient ) topology, since the set of
$g_{3}=e^{i\theta\sigma_{3}}$ does not form a closed normal subgroup. In this
case, two massless NG bosons may be expected but we find only one due to the
Nielsen-Chadha anomaly.

Let $G$ be a Lie group which gives the symmetry of a system, and its Lie
algebra as ${\bf g}={\rm Lie}(G)$. Let $S^{\alpha}$ ( $\alpha=1,\cdots,{\rm
dim}(G)-n_{SB}$ ) denote the generators ( a set of bases of ${\rm Lie}(G)$ )
correspond to remaining symmetries, and let $X^{\beta}$ (
$\beta=1,\cdots,n_{SB}$ ) imply the broken generators. From the orthogonality
condition of the bases of ${\bf g}={\rm Lie}(G)$, the Lie brackets
$[S^{\alpha},X^{\beta}]$ always belong to the linear space of broken
generators. While, any commutator of broken generators will be given by a
linear combination of all generators,

$\displaystyle[X^{\beta},X^{\gamma}]=\sum c^{\alpha}S^{\alpha}+\sum
c^{\delta}X^{\delta}.$ (7)

Hence, if the corresponding charges $Q^{S^{\alpha}}$ of $S^{\alpha}$ are
conserved and simultaneously they are Lorentz symmetric, and if the vacuum of
the theory is also Lorentz symmetric, then $Q^{S^{\alpha}}|0\rangle=0$ is
concluded immediately. On the contrary, $Q^{X^{\delta}}|0\rangle\neq 0$ ( for
$\forall\delta$ ) even if they are Lorentz scalar. If the breaking scheme
$G\to H$ ( $G$, $H$; Lie groups ) forms a coset $G/H$ and if it is a symmetric
space, then any Lie bracket of broken generators belongs to ${\rm Lie}(H)$
[20,32,43]:

$\displaystyle[X^{\alpha},X^{\beta}]\subset{\bf h}={\rm Lie}(H),\quad
S^{\alpha}\in{\bf h},\quad X^{\beta}\in{\bf m},\quad{\rm Lie}(G)={\bf g}={\bf
h}+{\bf m}.$ (8)

In that case, the VEV of any $[X^{\alpha},X^{\beta}]$ always vanishes in the
case of Lorentz symmetric conserved charges belong to ${\bf h}$. ( Therefore,
if the relation of Watanabe and Brauer is correct and is valid also in a
symmetric space, then the number of massive modes is given as a function of
VEVs of symmetric generators. ) It is interesting for us to consider several
models defined over Riemannian ( global ) symmetric spaces of the Cartan
classification [32]. Later, we will discuss how a local nature of anomalous NG
theorem is extended to a global structure in a symmetric space.

The equation of Lie algebra $[X^{\beta},X^{\gamma}]=\sum
c^{\alpha}S^{\alpha}+\sum c^{\delta}X^{\delta}$ means that the left hand side
of commutator is expanded by the linear form of the right hand side: Namely,
this formula counts the dimension of the linear space which the commutator
belongs to, and especially after taking a VEV of both side of this equation,
it gives a subspace which the VEV of commutator is described, such like a two-
dimensional surface in a three-dimensional space. In this sense, the VEV of
this equation is ”algebro-geometric.” After employing a method of
compactification suitable for a breaking scheme, a deformation theory and
moduli space for such an algebraic variety [30,31] could be introduced ( but
not always ). Especially in the symmetric space mentioned above, ${\rm
dim}[X^{\alpha},X^{\beta}]={\rm dim}{\rm Lie}(H)$. If the matrix $\langle
0|[X^{\alpha},X^{\beta}]|0\rangle$ is obtained from the second-order
derivative of an effective potential expanded by NG bosons associated with
broken generators, the matrix might contain a nonvanishing part, embedded in
the total part of the second-order derivative, which gives the finite mass
eigenvalues for the NG bosons. Namely, the dimension of the matrix of
nonvanishing part is the dimension of the linear space of massive NG bosons. (
In fact, the proof given in the paper of Watanabe and Murayama, Ref. [82], can
be interpreted as a calculation of basis set of the linear space which the
mass matrix of NG bosons belongs. ) Any type of non-vanishing VEV of $\sum
c^{\alpha}S^{\alpha}+\sum c^{\delta}X^{\delta}$ defines which pair
$[X^{\alpha},X^{\beta}]$ forms a non-vanishing matrix element. You should
notice that the rank of a matrix implies the dimension of a linearly-
independent components of a matrix. It should be mentioned that the dimension
of mass matrix is obtained after taking a VEV of the vacuum $|0\rangle$, and
thus at the moment one can say nothing about the mass matrix when one takes a
displacement from the vacuum $|0\rangle$ ( a mass matrix is given by
displacements of displacements ). It also should be examined how these
conditions of counting the dimension of massive modes in a mass matrix defined
in the linear space of Lie algebra has the validity, in a Higgs-type bosonic
model, an effective action of composite model like the Nambu$-$Jona-Lasinio (
NJL ) model or QCD, or in the case of Coleman-Weinberg mechanism [13]. In this
paper, we mainly consider a Goldstone-Higgs-type bosonic Lagrangian/theory to
investigate our anomalous NG theorem. Since a displacement caused by a Lie
group action to an order parameter ( where it is a composite or an elementary
field ) is given by an adjoint action of a Lie group, thus, our result should
be valid also in an NJL-type composite model. This is, of course, also the
case in a Coleman-Weinberg mechanism of symmetry breaking. Later, we examine a
Goldstone-Higgs type bosonic model of kaon condensation, while an examination
of NJL or QCD demands us further investigation as another paper, due to the
fact that those theories demand us some heavy calculations.

A lot of discussions use an $SU(2)$ model with two broken generators
$\sigma_{x}$, $\sigma_{y}$ while $\sigma_{z}$ is symmetric (
$(\sigma_{x},\sigma_{y},\sigma_{z})\in{\rm Lie}(SU(2))$ ). In this case,
$Q^{z}\propto\sigma_{z}$ can be regarded as an unbroken charge even though
this breaking scheme does not give a coset,

$\displaystyle[Q^{x},Q^{y}]=iQ^{z}\to{\rm symmetric}$ (9)

and thus the commutator always vanishes for a Lorentz-symmetric vacuum if
$Q^{z}$ is Lorentz-scalar and a conserved quantity. Note that only a symmetric
generator ( namely $Q^{z}$ ) appears in the right hand side in this expansion,
even though the breaking scheme does not give a coset ( quotient ) and of
course not a symmetric space: This breaking scheme is special from several
points. Moreover, if the following relations of VEVs holds,

$\displaystyle\langle[Q^{x},Q^{y}]\rangle=i\langle Q^{z}\rangle\neq 0,$ (10)
$\displaystyle\langle[Q^{x},Q^{z}]\rangle=-i\langle Q^{y}\rangle=0,$ (11)
$\displaystyle\langle[Q^{y},Q^{z}]\rangle=i\langle Q^{x}\rangle=0,$ (12)

then, they are isomorphic with the three-dimensional Heisenberg algebra,

$\displaystyle[x,y]=z,\qquad[x,z]=0,\qquad[y,z]=0.$ (13)

Thus, the insight of Nambu given in Ref. [62] which states that $Q^{x}$ and
$Q^{y}$ form a canonical conjugate pair in the case of ferromagnet $\langle
Q^{z}\rangle\neq 0$ is mathematically natural. Since a vacuum of the theory
must be chosen to evaluate VEVs of these brackets, the theory of $SU(2)$ is
expanded at the origin of the Lie group manifold by the NG bosonic
coordinates. Thus, the transformation from the Lie algebra to the Heisenberg
algebra is achieved at the origin of the Lie group and the corresponding
Heisenberg group. Physically, such a Heisenberg-algebra relation directly
concludes the Heisenberg uncertainty principle in the ”dynamical degrees of
freedom”, hence two NG-bosonic coordinates generated by the conserved charges
over a group manifold may acquire a quantum uncertainty. A comment on the
appearance of a symplectic vector space or group might be possible as some
literature already have done ( Refs. [81-84] ), though the notion and
structure of (quasi-)Heisenberg algebras/groups are better to emphasize the
quantum nature, since a Poisson bracket of classical mechanics also satisfies
a symplectic structure. In fact, a Heisenberg algebra is a central extension
of an algebra of symplectic vector space, and an automorphism of Heisenberg
algebra is given by a group of symplectic type. Such a symplectic vector space
of course defines a symplectic structure such like $\omega=\sum dp_{i}\wedge
dq_{i}$ up to an isomorphism, which is given by a so-called Lagrangian
subspace/submanifold. It should be emphasized that the transform from a Lie
algebra to a Heisenberg algebra under the prescription given above is not
achieved by some kind of perturbation or an analytic expansion ( such as a
deformation quantization of Poisson manifold [45] ) but by a functor, a
functorial manner provided by quantum field theory. It is a known fact that
both a three-dimensional Heisenberg group and $SU(2)$ can be embedded into
$SU(2,1)$ [46]. Thus, both the three-dimensional Heisenberg algebra and ${\rm
Lie}(SU(2))$ can be derived from the same Lie group, and it is interesting for
us to know how those ”submanifolds” are related with each other inside a
larger Lie group, and how an automorphism of ${\rm Lie}(SU(2))$ are related
with that of the Heisenberg algebra, vice versa. ( The groups of automorphisms
of $G$ and ${\rm Lie}(G)$ are isomorphic in general. Thus, the transform from
${\rm Lie}(SU(2))$ to the three-dimensional Heisenberg algebra may give a
global correspondence via their automorphism groups. ) Let us investigate this
problem by ourselves. ${\rm Lie}(SU(2,1))$ is an eight-dimensional algebra,
while ${\rm Lie}(SU(2))\simeq{\rm Lie}(SU(1,1))\simeq{\rm
Lie}(Sp(2))\simeq{\rm Lie}(SL(2,{\bf R}))\simeq{\rm Lie}(SL(2,{\bf C}))$ and
the three-dimensional Heisenberg algebra define three-dimensional linear
spaces. Thus, the three-dimensional spaces of ${\rm Lie}(SU(2))$ and the
Heisenberg algebra are embedded in the eight-dimensional space of ${\rm
Lie}(SU(2,1))$, and the anomalous NG theorem gives a mapping ( possibly a
bijection ) between two spaces. $SU(2)$ defines a sphere $S^{2}$, namely a
curve or a compact Riemann surface, and thus the corresponding three-
dimensional Heisenberg group should also define a curve or a Riemann surface.
Hence, the correspondence ( functor ) between ${\rm Lie}(SU(2))$ and the
three-dimensional Heisenberg algebra cause a correspondence between two curves
or Riemann surfaces with some globalization of those Lie algebras, especially
via exponential mappings. The NG bosons of the breaking scheme $SU(2)\to U(1)$
define a subset of $S^{2}$ ( a set of circles ). This implies that the NG
bosons of this breaking scheme gives a subspace of a Riemann surface, and thus
the three-dimensional Heisenberg algebra also gives a subspace of a Riemann
surface as the Heisenberg group manifold.

Let us examine the case of $SU(3)$ ( its Lie algebra is isomorphic with
Lie$SL(3,{\bf C})$ ). The definition of the Gell-Mann matrix representation of
${\rm Lie}(SU(3))$ is

$\displaystyle[Q^{A},Q^{B}]=if^{ABC}Q^{C},\quad Q^{C}\in{\rm
Lie}(SU(3)),\quad(A,B,C=1,2,\cdots,8)$ $\displaystyle f^{123}=1,\quad
f^{147}=f^{165}=f^{246}=f^{257}=f^{345}=f^{376}=\frac{1}{2},\quad
f^{458}=f^{678}=\frac{\sqrt{3}}{2}.$ (14)

For example, in the case of $\langle Q^{3}\rangle\neq 0$, $\langle
Q^{8}\rangle\neq 0$ with all of other generators have vanishing VEVs, the set
of following VEVs gives a ”quasi” Heisenberg algebra:

$\displaystyle\langle[Q^{1},Q^{2}]\rangle=i\langle
Q^{3}\rangle,\quad\langle[Q^{1},Q^{3}]\rangle=\langle[Q^{2},Q^{3}]\rangle=0,$
$\displaystyle\langle[Q^{4},Q^{5}]\rangle=\frac{i}{2}\langle
Q^{3}\rangle+\frac{i\sqrt{3}}{2}\langle Q^{8}\rangle,$
$\displaystyle\langle[Q^{4},Q^{3}]\rangle=\langle[Q^{5},Q^{3}]\rangle=\langle[Q^{4},Q^{8}]\rangle=\langle[Q^{5},Q^{8}]\rangle=0,$
$\displaystyle\langle[Q^{6},Q^{7}]\rangle=-\frac{i}{2}\langle
Q^{3}\rangle+\frac{i\sqrt{3}}{2}\langle Q^{8}\rangle,$
$\displaystyle\langle[Q^{6},Q^{3}]\rangle=\langle[Q^{7},Q^{3}]\rangle=\langle[Q^{6},Q^{8}]\rangle=\langle[Q^{7},Q^{8}]\rangle=0.$
(15)

The VEVs of all other brackets vanish and ”commute.” Strictly speaking, the
set of VEVs in this case does not give a Heisenberg algebra in the sense of
its definition given below, and we need to remove $Q^{3}$ or $Q^{8}$ from the
algebra to set $\langle Q^{3}\rangle=0$ or by $\langle Q^{8}\rangle=0$. While
we observe that a pairwise decoupling takes place. All of
$(Q^{1},Q^{2},Q^{4},Q^{5},Q^{6},Q^{7})$ are broken at the case $\langle
Q^{3}\rangle\neq 0$ and $\langle Q^{8}\rangle\neq 0$, or at the case $\langle
Q^{3}\rangle\neq 0$ and $\langle Q^{8}\rangle=0$ ( those cases give the scheme
$SU(3)\to U(1)\otimes U(1)$ ), while $(Q^{4},Q^{5},Q^{6},Q^{7})$ are broken
and $(Q^{1},Q^{2},Q^{3},Q^{8})$ remain unbroken ( this case gives the breaking
scheme $SU(3)\to SU(2)\otimes U(1)$ ) in the case $\langle Q^{3}\rangle=0$ and
$\langle Q^{8}\rangle\neq 0$. However, if we change the representation of
Lie$SU(3)$ from the Gell-Mann-type to others such as canonical basis, then we
find all generators except the Cartan subalgebra will be broken when we give a
finite VEV for one of elements of Cartan subalgebra of Lie$SU(3)$. Since a
physical phenomenon which depends on our choice of representation of a Lie
algebra never takes place in the nature, we conclude that the choice $\langle
Q^{3}\rangle=0$ and $\langle Q^{8}\rangle\neq 0$ is a special case of the
Gell-Mann representation, never occurs in the nature. Moreover, when we seek a
Heisenberg algebra in a Goldstone-type bosonic model of $SU(3)$, we need to
choose the form of VEV to make these charges broken when $\langle
Q^{3}\rangle\neq 0$ and $\langle Q^{8}\rangle=0$ ( the case $SU(3)\to
SU(2)\otimes U(1)$ mentioned above ). In such a case, the bosonic field $\Phi$
belongs to ${\bf 3}$-representation, and $\Phi^{\dagger}Q^{3}\Phi\neq 0$ and
$\Phi^{\dagger}Q^{8}\Phi=0$ gives an additional condition to the three
components of $\Phi$. Namely, we have to perform a variation of a subspace of
complex three ( real six ) dimensional space: This is not natural. Thus, we
conclude that the diagonal breaking of $SU(3)$ always gives not a Heisenberg
but a quasi-Heisenberg algebra. This fact indicates that the Heisenberg
uncertainty relation might be modified in quantum mechanical description of
fluctuations ( namely, NG bosons ) of the NG sector of the diagonal breaking
of $SU(3)$, such that,

$\displaystyle\Delta\chi^{1}\Delta\chi^{2}\geq
C^{a},\quad\Delta\chi^{4}\Delta\chi^{5}\geq C^{b},\cdots.$ (16)

Hence we speculate the deviation from the Heisenberg algebra given by a set of
VEVs of the Cartan subalgebra measures which pair of NG modes is more
”classical” and which pair of NG modes has a quantum fluctuation stronger than
others. A deviation from the Heisenberg-type uncertainty relation might affect
on quantum fluctuation in quantum phase transition.

The definition of Heisenberg algebra is

$\displaystyle[p_{i},q_{j}]=\delta_{ij}z,\quad[p_{i},z]=[q_{j},z]=0,$ (17)

where $(p_{1},\cdots,p_{n},q_{1},\cdots,q_{n},z)$ gives the generator of the
algebra. Thus, the number of generators must be odd in the Heisenberg algebra.
Here, $z$ is a central element of the Heisenberg algebra. Hence the finite
VEVs of elements of Cartan subalgebra give the center of the ( quasi )
Heisenberg algebra we have obtained, namely a central extension. A quasi-
Heisenberg algebra is defined to be

$\displaystyle[p_{i},q_{j}]=\delta_{ij}\sum_{\alpha}z_{\alpha},\quad[p_{i},z_{\alpha}]=[q_{j},z_{\alpha}]=0.$
(18)

Moreover, the expansion

$\displaystyle\langle[X^{\beta},X^{\gamma}]\rangle$ $\displaystyle=$
$\displaystyle\sum c^{\alpha}\langle S^{\alpha}\rangle+\sum c^{\delta}\langle
X^{\delta}\rangle$ (19)

must be pairwise decoupled to obtain a Heisenberg algebra: This is in general
not the case. Hence, to obtain a Heisenberg algebra via VEVs of Lie brackets
of generators of a Lie algebra, a subset of generators of odd number must give
a subalgebra. In this sense, an algebra generically obtained from the VEVs of
conserved charges should be called as a deformed/quasi Heisenberg algebra. The
uncertainty relation in a quasi-Heisenberg algebra should be investigated in
detail, since it might give a violation or a modification of the ordinary
Heisenberg uncertainty relation in a spontaneous symmetry breaking system, and
it would be confirmed experimentally from physical behaviors of NG sectors in
a condensed matter or a nucleus, especially in their quantum critical
phenomena [10].

Let us examine how a quasi-Heisenberg algebra arises by using the general
theory of Cartan decomposition, Cartan matrix, and canonical basis in the case
of semisimple Lie algebra Lie$(G)$ [20,32,43]. In this case, via the root
system ( so-called Cartan-Weyl basis ),

$\displaystyle{\bf g}$ $\displaystyle=$ $\displaystyle{\bf
h}\oplus\bigoplus_{\lambda\in R}{\bf g}_{\lambda},$ (20) $\displaystyle{\bf
g}_{\lambda}$ $\displaystyle=$ $\displaystyle\bigl{\\{}a\in{\bf
g}:[h_{j},a]=\lambda(h_{j})a,\quad\forall h_{j}\in{\bf h}\bigr{\\}},$ (21)

( $\lambda(h_{j})$: Cartan matrix, $R$: roots ), the Lie algebra is
generically defined by

$\displaystyle{\bf g}={\bf h}\oplus{\bf e}\oplus{\bf f},\quad
h_{i},h_{j}\in{\bf h},\quad e_{i},e_{j}\in{\bf e},\quad f_{j}\in{\bf f},$ (22)
$\displaystyle[h_{i},h_{j}]=0,$ (23)
$\displaystyle[e_{i},f_{j}]=\delta_{ij}h_{i},$ (24)
$\displaystyle[h_{i},e_{j}]=a_{ij}e_{j},$ (25)
$\displaystyle[h_{i},f_{j}]=-a_{ij}f_{j}.$ (26)

Here, $a_{ij}$ denote the Cartan matrix, and ${\bf h}$ is the Cartan
subalgebra. Thus, if $\langle e_{i}\rangle=\langle f_{j}\rangle=0$ ( $\forall
i,j$ ) while some of the bases of Cartan subalgebra take finite VEVs, $\langle
h_{j}\rangle\neq 0$, then a pairwise decoupling takes place and a quasi-
Heisenberg algebra is embedded in the total algebra. A Heisenberg pair is
given by the algebra basis of a positive and a negative roots. Thus, we obtain
the following theorem:

Theorem: Let $G$ be a semisimple Lie group and let Lie$(G)$ be its semisimple
Lie algebra. Let us assume the case where a theory only has VEVs of generators
toward the directions of Cartan subalgebra. Then the Cartan subgroup remains
unbroken, and the generators of Lie algebra will be pairwisely decoupled by
taking their VEVs, they form a quasi-Heisenberg algebra. This type of
decoupling never takes place in a Lorentz-invariant system due to the
vanishing condition $\langle Q\rangle=0$ of a conserved charge.

In such a situation of this theorem, a Lagrangian of NG bosons may be pairwise
decomposed inside the linear space of NG bosons at least in the quadratic part
of the Lagrangian of NG boson fields ( we discuss how and when such a
decomposition takes place in the NG sector of a theory in sec. 5 ). In the
case of breaking scheme $G\to$ Cartan subgroup, where all group elements
except the Cartan subgroup are broken, and all generators of Cartan subalgebra
take non-vanishing VEVs, then one can count the number of pairs ( namely, the
number of mode-mode couplings of NG bosons ) which give a quasi-Heisenberg
algebra:

$\displaystyle n_{pair}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Bigl{[}{\rm dim}{\rm Lie}(G)-{\rm rank}{\rm
Lie}(G)\Bigr{]}.$ (27)

Note that the rank of Lie$(G)$ coincides with the dimension of Cartan
subalgebra. If $G\to H$ gives a symmetric space $G/H$, then

$\displaystyle{\rm rank}{\rm Lie}(G)$ $\displaystyle=$ $\displaystyle{\rm
dim}{\rm Lie}(H)={\rm dim}[X^{\alpha},X^{\beta}]$ (28)

holds. Thus,

$\displaystyle n^{symmetric-space}_{pair}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Bigl{[}{\rm dim}{\rm Lie}(G)-{\rm dim}{\rm
Lie}(H)\Bigr{]}$ (29) $\displaystyle=$ $\displaystyle\frac{1}{2}\Bigl{[}{\rm
dim}{\rm Lie}(G)-{\rm dim}[X^{\alpha},X^{\beta}]\Bigr{]}.$

Moreover, if the Lagrangian of any pair of NG bosons is given in the form as
only one mode of a pair is massive by the mixing of modes inside the pair (
this situation will be given by an NG boson Lagrangian in sec. 5 ), then the
number of massive NG bosons equals the number of pairs, and

$\displaystyle n_{BS}$ $\displaystyle=$ $\displaystyle
n_{NG}+\frac{n_{pair}}{2}=n_{NG}+\frac{1}{2}\Bigl{[}{\rm dim}{\rm Lie}(G)-{\rm
rank}{\rm Lie}(G)\Bigr{]}.$ (30)

Since $[X^{\alpha},X^{\beta}]=\sum c^{\gamma}S^{\gamma}+\sum
c^{\delta}X^{\delta}$,

$\displaystyle{\rm dim}\langle[X^{\alpha},X^{\beta}]\rangle={\rm
dim}\Bigl{(}\sum c^{\gamma}\langle S^{\gamma}\rangle\Bigr{)}={\rm rank}{\rm
Lie}(G)$ (31)

holds in the diagonal breaking case. Hence we get the following equation for a
diagonal breaking:

$\displaystyle n_{BS}$ $\displaystyle=$ $\displaystyle
n_{NG}+\frac{1}{2}\Bigl{[}{\rm dim}{\rm Lie}(G)-{\rm
dim}\langle[X^{\alpha},X^{\beta}]\rangle\Bigr{]}.$ (32)

Here, ${\rm dim}{\rm Lie}(G)-{\rm dim}\langle[X^{\alpha},X^{\beta}]\rangle$
gives the number of Heisenberg pairs in the diagonal breaking scheme. Since
$n_{BS}=$dimLie$(G)-$dimLie$(H)$, we get

$\displaystyle n_{NG}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\Bigl{[}{\rm
dim}{\rm Lie}(G)-{\rm rank}{\rm Lie}(G)\Bigr{]}.$ (33)

More general case is examined by utilizing (20)-(26). When $\langle
h\rangle\neq 0$, $\langle e\rangle=\langle f\rangle=0$ ( a diagonal breaking
), then $e$ and $f$ are broken since $[\Phi,e]\neq 0$, $[\Phi,f]\neq 0$,
$\Phi\in h$. If $\langle h\rangle\neq 0$, $\langle e\rangle\neq 0$, $\langle
f\rangle=0$, then $h$, $e$, $f$ are broken since $[\Phi,h]\neq 0$,
$[\Phi,e]\neq 0$, $[\Phi,f]\neq 0$, $\Phi\in h\oplus e$. In the latter case,
the algebra of VEVs is not a (quasi) Heisenberg-type. An investigation on the
case of all generators are broken, $G\to$ nothing, becomes complicated to give
a general theory.

Since a Heisenberg group and its Lie algebra are realized on a symplectic
vector space, one can introduce a Darboux basis of a symplectic vector space,
corresponds to the canonical coordinates, to express the Heisenberg algebra.
Then the Heisenberg algebra acquires a geometric implication. Moreover, one
can introduce an operator algebra analysis of the anomalous NG theorem via the
Stone-von Neumann theorem [75]. Therefore, a unification of algebra, analysis,
and geometry takes place in our anomalous NG theorem. The Heisenberg algebra
$(p,q,z)$ obtained from our $SU(2)$ model is a special example of,

$\displaystyle X\stackrel{{\scriptstyle
p}}{{\rightarrow}}Y\stackrel{{\scriptstyle q}}{{\rightarrow}}Z,\quad
X\stackrel{{\scriptstyle q}}{{\rightarrow}}Y^{\prime}\stackrel{{\scriptstyle
p}}{{\rightarrow}}Z^{\prime},\quad Z\neq Z^{\prime}.$ (34)

Here, $X,Y,Z,Y^{\prime},Z^{\prime}$ implies some mathematical sets, and we
regard the canonical pair $(p,q)$ is given by certain types of morphisms. The
center $z$ given by the VEV $\langle Q^{z}\rangle$ in the $SU(2)\to U(1)$ case
measures how $Z$ and $Z^{\prime}$ are different. This kind of noncommutativity
appears in algebras of monodromy, holonomy, etc. This is the geometric nature
of the Heisenberg algebra as the essence of quantum mechanics. From the aspect
of Heisenberg groups, our Heisenberg algebra coming from the anomalous NG
theorem of an $SU(2)$ model can be related with the three-dimensional compact
Iwasawa manifold obtained from $\Gamma\backslash G/H$ of the three-dimensional
Heisenberg group $G$, where

$\displaystyle G$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}1&a&b\\\ 0&1&c\\\
0&0&1\end{array}\right),\quad a,b,c\in{\bf R},$ (38) $\displaystyle H$
$\displaystyle=$ $\displaystyle\\{e\\},$ (39) $\displaystyle\Gamma$
$\displaystyle=$ $\displaystyle G\cap GL(3,{\bf Z}).$ (40)

It is a known fact that a complex structure is found in a three-dimensional
Iwasawa manifold [40]. From the Kodaira-Spencer theory of deformation of
complex structure of a complex manifold [44], we know the obstruction of
deformation of a complex structure is determined by the cohomology group of
the manifold, namely it is one of global aspects/structures of the NG manifold
of our anomalous NG theorem.

An interesting fact is that a three-dimensional Heisenberg algebra $[p,q]=z$,
$[p,z]=[q,z]=0$ is constructed by the differential operators such as
$p=\partial_{1}-\frac{x^{2}}{2}\partial_{3}$,
$q=\partial_{2}+\frac{x^{1}}{2}\partial_{3}$, $z=\partial_{3}$: In that case,
$(p,q,z)$ forms an orthogonal frame of an appropriate manifold. It is
interesting for us to compare this fact with the representation of
differential operator expression of the $sl_{2}$-triple. It is a well-known
fact that a Heisenberg algebra can be expressed by a Weyl algebra,

$\displaystyle[x_{i},\partial_{j}]=-\delta_{ij},\quad[x_{i},x_{j}]=[\partial_{i},\partial_{j}]=0,$
(41)

and the Weyl-algebra expression of our quasi-Heisenberg algebra gives us a
further implication of mathematical and geometric nature of our Lorentz-
violating NG boson Lagrangian ( see sec. 5 ). A Weyl algebra is a simple
Nötherian integral domain, and it has a global dimension $n$. It should be
mentioned that any term higher than the second-order of the expansion of an
adjoint action expressed by an exponential mapping, a similarity
transformation $g^{-1}Qg$ ( $Q$; a conserved charge ), or in a Killing form (
kinetic term ) of the Lagrangian or the effective potential, are fall into the
fundamental relation of quasi-Heisenberg algebra (18) after taking their VEVs,
and thus those expansions are effectively ”terminated” at the quasi-Heisenberg
algebra ( such a truncation can take place, of course, in a quantum theory ),
and thus, only the subset of (quasi-)Weyl algebra of universal enveloping
algebra appears in a theory: This case is coming from the fact that our
theoretical framework is suitable in the vicinity of the ground state of the
system determined by a choice of the form of VEVs and we consider a Heisenberg
algebra, not a Heisenberg group. The Weyl algebra itself is isomorphic with
the Moyal algebra of deformation quantization, thus the global character of
our theory of anomalous NG theorem will acquire a connection with the Moyal-
Weyl deformation quantization. Moreover, a Weyl algebra defines several
differential operators which directly connects with theory of $D$-modules [4].
A Weyl algebra of an infinite order, possibly isomorphic with a deformation
quantization, will be entered into our anomalous NG theorem when we consider
the corresponding Heisenberg group: Naively, the Weyl algebra is terminated at
the order of the elementary relations ( Lie brackets, (38) ) of the
corresponding quasi-Heisenberg algebra, as we have stated, while a linear
transformation of a basis of representation space of the quasi-Heisenberg
algebra caused by an operation of Heisenberg group ( an adjoint action to the
Heisenberg algebra ) gives the Weyl algebra which can acquire its higher-order
products and derivatives of algebras expanded by the set
$(x_{j},\partial_{j})$. To make this matter consistent, we need to obtain the
notion of quasi-Heisenberg group. It should be investigated that a Moyal-Weyl
type deformation quantization for the quasi-Heisenberg/quasi-Weyl algebra
starting from a Poisson manifold defined by a Poisson structure of broken
generators, which might give some results confirmed by experiments. From our
observation of $SU(3)$, we can propose the following modified Moyal-Weyl
product:

$\displaystyle f*g$ $\displaystyle=$ $\displaystyle
f\exp\Bigg{(}(\sum_{j}\langle
h_{j}\rangle)\sum_{A,B}\frac{\overleftarrow{\partial}}{\partial\chi^{A}}\frac{\overrightarrow{\partial}}{\partial\chi^{B}}-\frac{\overleftarrow{\partial}}{\partial\chi^{B}}\frac{\overrightarrow{\partial}}{\partial\chi^{A}}\Bigg{)}g$
(42) $\displaystyle=$ $\displaystyle fg+\sum_{j}\langle
h_{j}\rangle\\{f,g\\}_{PB}+\cdots,$ $\displaystyle\\{f,g\\}_{PB}$
$\displaystyle=$ $\displaystyle\frac{\partial
f}{\partial\chi^{A}}\frac{\partial g}{\partial\chi^{B}}-\frac{\partial
f}{\partial\chi^{B}}\frac{\partial g}{\partial\chi^{A}}$ (43)

This modified Moyal-Weyl product provides an example of a multi-
Planck-”constant” model which has been proposed by the author in Ref. [70].
The set of VEVs of the Cartan subalgebra generators gives several Planck
”constants” which depend on a position of the Lie group manifold or a local
geometry, mainly curvatures of $V_{eff}$. ( An example of deformation
quantization of Heisenberg manifolds, see [73]. ) It should be mentioned that
our approach to the anomalous NG theorem heavily depends on Lie algebras of
conserved charges and they are the special cases of current algebras, for
example, $[j^{A}(x_{0},{\bf x}),j^{B}(x_{0},{\bf
y})]=if^{ABC}\delta^{(3)}({\bf x}-{\bf y})j^{C}(x)$. Hence, our result may be
reformulated by those current algebras and their globalizations ( it may be
called as ”current groups”, and they might introduce new mathematics in our NG
theorem, especially from the context of integral geometry and operator
algebras ). In Ref [46], a quasiconformal mapping of a Heisenberg group is
studied. While, a quasiconformal mapping is obtained in a Moyal-Weyl
deformation quatization [67]. It is interesting for us to unify these
approaches.

In more generic case, a Lie algebra is defined as a direct sum of finite
number of Lie subalgebras:

$\displaystyle{\bf g}$ $\displaystyle=$ $\displaystyle{\bf g}_{1}+\cdots+{\bf
g}_{l},$ (44)

and each of them has a root space decomposition. Thus the theorem given above
can be stated differently:

Theorem: Any diagonal breaking which remains the Cartan subalgebra unbroken
converts the Lie algebra into a direct sum of a finite number of quasi-
Heisenberg algebras and Abelian algebras, via taking the VEVs of algebra
generators. Turn to the group theory, such a breaking scheme gives a direct
product of quasi-Heisenberg groups and Abelian Lie groups via the anomalous NG
theorem in quantum field theory. The quasi-Heisenberg group and Abelianized
group act on the effective action/potential of the theory and its low energy
effective theory.

Since this paper considers internal symmetries mainly for our anomalous NG
theorem, we choose a semisimple Lie group such as $SU(n)$, $SO(n)$ and $Sp(n)$
as the main subject, though we can examine more generic cases of $SL(n,F)$ and
$GL(n,F)$ ( $F$: a number field of characteristic zero ). Note that any Lie
group has an associated Lie algebra. Moreover, especially ${\rm Lie}(SL(n,F))$
and ${\rm Lie}(GL(n,F))$ are finite dimensional, we have a functor which gives
corresponding simply-connected Lie groups. Hence our analysis presented here
is valid to those Lie algebras. The Ado-Iwasawa theorem states that a finite-
dimensional Lie algebra defined over a field $F$ has a faithful finite-
dimensional representation. A general consideration of some cases of ${\rm
Lie}(SL(n,F))$ and ${\rm Lie}(GL(n,F))$ contains theories of $SU(N)$, $SO(N)$,
…, as their special examples. Especially, $GL(2,{\bf R})$ is disconnected into
two parts according to the signature $\pm 1$ of determinant, thus a problem of
covering on the space of NG bosons, a problem of the global nature of the NG
theorem similar to the case of gauge orbits, gives an interesting subject
which has a strong connection with number theory. Though, for examining the
mechanism to generate massive NG bosons, group elements in the vicinity of
identity are important since we examine small fluctuations of bosonic fields
from a stationary point. It is a known fact that if a Lie group is simply
connected, its global structure is determined by the corresponding Lie
algebra. The global structure of the effective action/potential of a theory
will be known by both its group theoretical nature and quantum field theory.
In a global aspect, $Spin(N)$ is the double-covering group of $SO(N)$, and
thus, if $SO(N)$ gauge theory has a unique vacuum in its $SO(N)$ fundamental
domain, the corresponding $Spin(N)$-gauge theory must have two exactly
degenerate vacua, though they cannot be distinguished by the Lie algebra (
namely, a locally defined quantity ) in general.

Our anomalous NG theorem has not only a locally characteristic aspect ( quasi-
Heisenberg algebra, massive NG bosons, Weyl algebra ) but also some
interesting global nature, which reflect some number theoretical aspects such
as a fundamental group of covering group or a Galois group. Such a global
aspect of our anomalous NG theorem directly reflects to symmetry between
stationary points and geometry of stationary points, given by the effective
potential $V_{eff}$. For our understanding of the global nature of an NG
manifold/variety ( probably some class of quotients of breaking schema, if it
has a coset topology, may have fixed points and singularities ) of the
anomalous NG theorem, we need further investigation. Some exotic mathematical
nature of NG manifolds provides us a new subject of study on submanifolds in (
differential ) geometry. The global character of an NG manifold, namely its
compactness or the fundamental group, is understood by neglecting the local
details of effective potential, and then convert our problem to a problem of
Lie groups and homogeneous spaces. Sometimes $G\to H$ gives a
Riemannian/Hermitian symmetric space, while it might be possible to generate a
pseudo-Riemannian space as a result of symmetry breaking ( this is not
familiar from a context of physics ). Topological/global nature of pseudo-
Riemannian spaces is still not yet understood enough in modern mathematics
[43]. Since there is a common understanding that a fundamental group is a
Galois group [72], our problem continues to the region of number theory. From
this aspect of the global character of geometry of symmetry breaking,
especially the Clifford-Klein form $\Gamma\backslash G/H$ is important since
it has some nice properties, and the case where $\Gamma$ is proper
discontinuous and free is interesting for us. If a breaking scheme $G\to H$
gives a Hermitian symmetric space $G/H$, then it is a known fact that the
$G/H$ has a uniform lattice $\Gamma$ ( i.e., $\Gamma\backslash G/H$ is compact
).

A more complicated situation will arise when we consider a breaking scheme
under some generators of a Lie algebra are already broken by an explicit
symmetry breaking parameter, namely so-called ”explicit+spontaneous” symmetry
breakings [70]. ( We give an example ( the kaon condensation model ) of it (
anomalous+explicit+spontaneous symmetry breaking ) later in this paper. ) In
such situations, for example, a breaking scheme such as

$\displaystyle SU(N)\to({\bf Z}/N{\bf Z})^{\times}\simeq Gal({\bf
Q}(\zeta_{N})/{\bf Q})$ (45)

can take place. Here $\zeta_{N}$ is the $N$-th root of unity, and $({\bf
Z}/N{\bf Z})^{\times}$ gives the center of $SU(N)$. ( Note that $\det[{\rm
diag}(\underbrace{\zeta_{N},\cdots,\zeta_{N}}_{N})]=1$. ) Namely, it gives the
following central extension:

$\displaystyle 1\to({\bf Z}/N{\bf Z})^{\times}\to SU(N)\to PSU(N)\to 1.$ (46)

( $PSU(N)$; projective special unitary group. ) $Gal({\bf Q}(\zeta_{N})/{\bf
Q})$ is the Galois group of the cyclotomic extension. This exact sequence is
useful to consider a breaking scheme which gives a Grassmannian
$SU(N)/SU(N-M)SU(M)$ and then successively ${\bf Z}/(N-M){\bf Z}\times{\bf
Z}/N{\bf Z}$. This breaking scheme may have a quite interesting mathematical
implication in a quantum group by utilizing the quasi-Heisenberg and quasi-
Weyl algebra representations: The discrete Heisenberg group ( all of the
matrix elements of representation of a discrete Heisenberg group are integers
) is given by the algebraic relations of generators such that $xy=zyx$,
$[x,z]=[y,z]=0$. The author speculate this is the first time to find a quantum
algebra in the NG theorem. Needless to say, the relation $xy=zyx$ is
consistent with the canonical Heisenberg algebra $[x,y]=1$ by setting
$xy=z/(z-1)$ and $yx=1/(z-1)$ ( they recover the canonical commutation
relation ). Moreover, Bost and Connes construct a theory of dynamical system
which gives a Galois group $Gal({\bf Q}(\zeta_{N})/{\bf Q})$ associated with a
spontaneous symmetry breaking, and the partition function below the critical
temperature is the Riemann zeta function [7]. It is emphasized that their work
has a strong connection with the Riemann hypothesis. In fact, our theory of NG
theorem contains some parts of algebraic aspects of their work naturally. For
example, a global nature of our anomalous NG theorem can give $Gal({\bf
Q}(\zeta_{N})/{\bf Q})$. Hence our theory of NG theorem might provide an
approach toward the solution of the Riemann hypothesis: This point will be
discussed later in this paper. We need a systematic investigation on the
relation between several symmetry-breaking schema and Galois representations,
with a perspective of (non)commutative class field theory, i.e., the Langlands
conjecture [22,23,24,29,58,80]. From similar perspective, the following short
exact sequences are also interesting:

$\displaystyle 1\to{\bf Z}/2{\bf Z}\to{\rm Spin}(N)\to SO(N)\to 1,$ (47)
$\displaystyle 1\to{\bf Z}/2{\bf Z}\to{\rm Spin}^{\bf C}(N)\to SO(N)\otimes
U(1)\to 1.$ (48)

In fact, ${\rm Spin}(N)$ is a double covering group of $SO(N)$, and ${\rm
Spin}^{\bf C}(N)$ is its complexification. Also, ${\rm Spin}(N,{\bf R})$ is
the group in the theory of Clifford algebra. ${\bf Z}/2{\bf Z}$ is a Galois
group. The upper exact sequence is frequently used for an explanation of a
Stiefel-Whitney class which judges whether a manifold is orientable. A central
extension of Lie group by a discrete group corresponds to the covering space,
directly related with the fundamental group. Moreover, a central extension of
Lie group induces a central extension of Lie algebra ( but its inverse is not
true in general ). The Lie’s third theorem states that a simply connected Lie
group exists for a given finite dimensional Lie algebra. We make a brief
comment on a central extension of Lie algebra [63]. For example,

$\displaystyle 0\to{\bf R}\to\widetilde{{\rm Lie}(G)}\to{\rm Lie}(G)\to 0.$
(49)

Here, $\widetilde{{\rm Lie}(G)}$ is the central extension of ${\rm Lie}(G)$ by
${\bf R}$. Some literature given as our references discuss possible roles of
central extensions to Lie brackets which might affect on the anomalous
behavior of NG theorem. It is a well-known fact that there is no nontrivial
central extension if ${\rm Lie}(G)$ is semisimple. A central extension may
have a role when we consider a symmetry of a Kac-Moody group or a Heisenberg
group. The following isomorphism is useful for us:
$\pi_{1}(G/H)\simeq\pi_{1}(G)$ where $G$ is a connected Lie group, and $H$ is
a simply connected closed subgroup of $G$. Thus, the nature of covering space
which is implied by a central extension of a Lie group conserves under a
breaking scheme $G\to H$, and it is enough for us to consider a
fundamental/Galois group of covering space of $G$. Those covering groups and
Galois groups describe symmetries of stationary points of NG sectors. For
example, the set of stationary points inside the fundamental domain of $G/H$
acquires the symmetry of $\pi_{1}(G/H)$.

From our examination, there are functors of algebra cohomologies associated
with a breaking scheme of anomalous NG theorem, such that,

$\displaystyle{\rm Lie\,algebra\,cohomology}\to{\rm
Heisenberg\,algebra\,cohomology}\to{\rm Galois\,cohomology}.$ (50)

This is a remarkable fact since, for example, a Lie algebra cohomology
describes the topological nature of underlying Lie group. If a symmetry of a
set of stationary points in an NG sector is a Galois type, the set gives a
Galois representation controlled by a Galois cohomolgy. These cohomology,
especially a Galois cohomology may have an overlap with an étale cohomology
since a Galois cohomology is a special case of étale cohomology which implies
an underlying algebraic variety. This fact may help us to understand the
underlying mechanism of the relations of those cohomologies and algebras. The
relationship between a Heisenberg algebra and a Galois group is a
characteristic aspect of our anomalous NG theorem, while other relations may
be contained also in the normal NG theorem. A Galois group appears in various
geometric examples but of particular interest here is geometric expressions of
class field theory, several Galois representations, and étale fundamental
groups by our anomalous NG theorem. It may be noteworthy to mention that the
Abelianized part of the total Lie algebra reflects the flatness of the
effective action/potential of the theory, while the quasi-Heisenberg relation
lifts partly the degeneracy of the vacua of the theory along with some NG-
bosonic coordinates. Since an apparent discrete symmetry between stationary
points takes place in a massive NG-bosonic coordinate/space, a Galois
representation will be found in the space of a quasi-Heisenberg relation.

Now, we list some breaking schema interesting for us from the context of this
paper:

$\displaystyle SU(4)\to SU(2)\otimes SU(2)\simeq SO(4)\to SU(2)_{diag}\to
U(1),$ (51) $\displaystyle SU(5)\to SU(3)\otimes SU(2)\otimes U(1)\to
U(1)\otimes U(1)\otimes U(1)\otimes U(1),$ (52) $\displaystyle SU(6)\to
SU(3)_{L}\otimes SU(3)_{R}\to SU(3)_{V}\to U(1)\otimes U(1),$ (53)
$\displaystyle SO(10)\to SU(4)\otimes SU(2)\otimes SU(2),$ (54) $\displaystyle
E_{6}\to SO(10)\otimes U(1)\to SU(5),$ (55) $\displaystyle E_{8}-{\rm
ferromagnet},\quad({\rm experimentally\,observed\,spin\,system}),$ (56)
$\displaystyle G_{2}\to SO(4),$ (57) $\displaystyle SU(N)\to
SU(N-M),\quad({\rm Stiefel\,manifold}),$ (58) $\displaystyle SO(N)\to
SO(N-M),\quad({\rm Stiefel\,manifold}),$ (59) $\displaystyle SU(N)\to
SU(N-M)\otimes SU(M),\quad({\rm symmetric\,space,\,Grassmann}),$ (60)
$\displaystyle SO(N)\to SO(N-M)\otimes SO(M),\quad({\rm
symmetric\,space,\,Grassmann}),$ (61) $\displaystyle Spin(6)=SU(4)\to
something,$ (62) $\displaystyle Spin(4,2)=SU(2,2)\to something.$ (63)

In those examples, if a symmetry breaking takes place under a breaking scheme
in which some elements of the Cartan subgroup of the total group remains
unbroken, then it is trivial that a ( quasi ) Heisenberg algebra arises. For
example, the breaking scheme $SU(N)\to SU(N-M)\otimes SU(M)$ will take place
by an order parameter of the form
diag$(\underbrace{a,\cdots,a}_{N-M},\underbrace{b,\cdots,b}_{M})$ which should
be proportional to a linear combination of VEVs of the Cartan subalgebra of
$SU(N)$. A large part of breaking schema listed above are fall into this class
of symmetry breakings. The breaking scheme $SU(N)_{L}\otimes SU(N)_{R}\to
SU(N)_{V}$ is famous in a chiral symmetry breaking of left-right symmetric
$N$-flavor model. In this case, the Lie algebra one considers is

$\displaystyle[\theta^{a}T^{a}\otimes
1+\varphi^{b}T^{b}\otimes\sigma^{3},\Phi],$
$\displaystyle\Phi\propto\sigma^{1},\quad\theta^{a}T^{a}\otimes 1\in{\rm
Lie}(SU(N)_{V}),$
$\displaystyle(\theta^{a}+\varphi^{a})T^{a}\otimes\frac{1+\sigma^{3}}{2}\in{\rm
Lie}(SU(N)_{L}),$
$\displaystyle(\theta^{a}-\varphi^{a})T^{a}\otimes\frac{1-\sigma^{3}}{2}\in{\rm
Lie}(SU(N)_{R}).$ (64)

Here, $\theta^{a}T^{a}\otimes 1$ remains symmetric while
$\varphi^{b}T^{b}\otimes\sigma^{3}$ is broken. Hence, the VEV takes its value
toward $\sigma^{1}$ direction ( as you know, $\gamma^{0}$ is frequently used,
while $\sigma^{3}\to\gamma_{5}$ ), and this case does not belong to the class
of diagonal breaking we have studied in this paper.

$E_{8}$ might have several exotic breaking schema due to its Dynkin diagram
and Cartan martrix while our observation of anomalous NG theorem in a generic
case should valid to it. An example of spin system of $E_{8}$ symmetry has
been observed experimentally quite recently [6,9,88]. Zamolodchikov seems to
use his theory ( affine Toda field theory of $E_{8}$ ) to give a mass spectrum
of mesons, two quark ( kink ) bound states and thus his theory is constructed
in a (1+1)-dimensional model, though some part of the mechanism of generating
an $E_{8}$ spectrum is independent from the dimensionality of a system,
determined by the Lie algebra Lie$(E_{8})$. Hence an $E_{8}$ spin system has
an importance from its own right, beyond its dimensionality. ( A breaking
scheme of $E_{8}$ is also interesting for us from the context of the Kazhdan-
Lusztig-Vogan polynomials for $E_{8}$ [38,39,51]. )

###  2.1 Riemannian and Hermitian Symmetric Spaces

If a breaking scheme $G\to H$ gives a symmetric space [32], several geometric
properties will be introduces to our NG theorem more concretely. Especially, a
local nature ( Lie algebra ) and a global structure ( Lie group ) is bridged
more clearly. First, we summarize the basic well-known fact of a symmetric
space. Let $M$ be a symmetric space. A Lie group $G$ acts transitively on $M$.
In addition, an involution $s$ is defined for any local point of $M$, as an
automorphism of $M$, and $s$ acts on any group element of $G$ as an adjoint
$sgs^{-1}$. This involution is an automorphism of $G$ itself, and of course it
acts on ${\rm Lie}(G)$ as an automorphism. Then ${\bf g}={\rm Lie}(G)$ is
decomposed into ${\bf h}+{\bf m}$ by their eigenvalues of operations of $s$ (
${\bf h}\to+1$, the Cartan subalgebra, and ${\bf m}\to-1$ ). Hence, by the
number of odd elements ${\bf m}$, the relations $[{\bf h},{\bf h}]\subset{\bf
h}$, $[{\bf h},{\bf m}]\subset{\bf m}$, $[{\bf m},{\bf m}]\subset{\bf h}$ are
immediately obtained ( this is a kind of grading of the algebra by the set of
odd elements ${\bf m}$ ). ${\bf m}$, an orthogonal complement space of ${\bf
h}$, is isomorphic with $TM$, and a curve $t\to e^{it{\bf m}}\cdot o$ ( $o$: a
point of $M$, $t\in{\bf R}$ ) is geodesic. As stated above, the tangent bundle
of a Riemannian manifold has $O(n)$ as the structure group. If a spontaneous
symmetry breaking $G\to K$ gives a symmetric space, $M=G/K=e^{i{\bf m}}$, and
the Cartan subalgebra remains unbroken, then the NG manifold is expanded only
by the basis of broken generators ${\bf m}$ which is isomorphic with
$TM=T(G/K)$ which may have the structure group $O(n)$, and the NG bosons
$\\{\chi^{A}\\}$ as the local coordinate system expressed by $\Phi\to
e^{i\chi^{A}m^{A}}\Phi$ are all geodesic, whether the normal or anomalous
cases of NG theorem. The VEVs $\langle{\bf h}\rangle\neq 0$ are always normal
( vertical ) with the tangent space given by the NG boson space ${\bf m}$. A
quasi-Heisenberg algebra is globally defined, inside the linear space $TM$.
Since an NG boson gives a geodesic over the manifold $G$, a Jacobi field is
associated along with the geodesic curve. The curvature tensor is given by
$R(X,Y)Z=[[X,Y],Z]\subset{\bf m}$ ( here, $X,Y,Z\in{\bf m}$ and we have used
the physics convention of $g=e^{i{\bf m}}$, where $i=\sqrt{-1}$ ) at the
origin. If an order parameter $\Phi$ belongs to ${\bf h}$, then the Riemann
curvature $R$ appears at the fourth order displacement of a Lagrangian or an
effective potential caused by broken generators. Here, a symmetric space is
defined by a Lie algebra, thus it does not depend on details of the manifold.
For example, $SU(N)\to SU(N-M)\otimes SU(M)$ is a symmetric space, thus broken
generators form the algebra $[X^{a},X^{b}]\subset S^{c}$, and if $\langle
S^{c}\rangle\neq 0$, a ( quasi ) Heisenberg algebra arises. In this case,
Lie$SU(N)$ is projected to the quasi-Heisenberg algebra globally via our
anomalous NG theorem. Especially interesting for us is the fact that a unitary
group $U(N)$ is a Hermitian manifold ( keeps a Hermitian structure of a
quadratic form ), while a Heisenberg group is possibly be described by a
Riemannian manifold. Hence, our anomalous NG theorem may bridge between a
complex ( Kähler ) structure and a quantized symplectic ( quasi-Heisenberg )
structure in a symmetric space.

###  2.2 A Heisenberg Group as the Symmetry of the Beginning

If a Heisenberg group ( sometimes used in a flavor dynamics, a flavor-symmetry
breaking ) is an internal symmetry from the beginning of a theory, and if one
considers its spontaneous symmetry breaking, a situation similar with the
cases of semisimple classical Lie groups takes place, since a set of VEVs of a
Heisenberg algebra can again give a Heisenberg algebra. For example,

$\Xi=\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&0\\\
0&0&0\end{array}\right)\quad{\rm or}\quad\left(\begin{array}[]{ccc}0&1&0\\\
0&0&0\\\ 0&0&0\end{array}\right)\quad{\rm
or}\quad\left(\begin{array}[]{ccc}0&0&1\\\ 0&0&0\\\ 0&0&0\end{array}\right),$
(65)

with an action of a Heisenberg group $G$ ( see, (35) ) from the left side
gives $G\Xi=\Xi$, namely $\Xi$ is $G$-singlet, can be utilized to make an
invariant theory. This form of $\Xi$ may be attractive for an attempt to
generate a flavor degree of freedom: For example, the following
$\widetilde{\Xi}$ which breaks a Heisenberg-group symmetry can generate a
flavor hierarchy by an action of $G$:

$\widetilde{\Xi}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&\epsilon&0\\\
0&0&\epsilon^{2}\end{array}\right),\quad|\epsilon|\ll 1.$ (66)

Therefore, one can consider a symmetry breaking which generates a flavor
hierarchy via our anomalous NG theorem:

SU(2) ( SU(N) ) $\to$ Heisenberg group $\to$ something.

An important issue is coming from the fact that the Killing form of a
nilpotent Lie algebra is identically zero, while the Killing form of a
solvable Lie group gives a different result from it. A quotient of Heisenberg
group gives a solvmanifold [3]. A fermion model of solvable Lie group symmetry
with a dynamical symmetry breaking which generates a sigma model might be
possible. In a bosonic field theory of a compact Lie group, the Killing form
is negative definite, while in the case of indefinite signature of Killing
form, a model Lagrangian should be regarded as an ”analytic continuation” from
a physical model.

###  2.3 Kac-Moody Algebras, Generalized Kac-Moody Algebras, and Affine Lie
Algebras

It should be noticed that a Lie group which derives a Nöther charge can be
treated ( at least, formally in a certain sense ) as an internal symmetry of a
model Lagrangian. We assume a Kac-Moody Lie algebra [25,37] has its
corresponding Lie group via an exponential mapping, surjectively, even though
this assumption is sometimes violated, and it is a non-trivial issue to define
a Haar measure for a Kac-Moody group. This issue affects to define a path-
integral measure over a Kac-Moody group. From our perspective of this paper,
the most interesting fact is that, after taking VEVs of brackets of a Kac-
Moody algebra ( now we have an infinite number of conserved charges ),
especially an affine Lie algebra, we obtain a direct sum of quasi-Heisenberg
algebras of polynomial growths ( would be called as an infinite-dimensional
quasi-Heisenberg algebra ) and Abelianized subalgebras. The method to
construct a Kac-Moody algebra ( a Cartan subalgebra, a root system, a Cartan
matrix, etc. ) is parallel with the general theory of finite-dimensional
simple Lie algebras [37]: Hence, our discussion given above can
straightforwardly be applied to several cases of Kac-Moody algebras. Our
theory can be regarded as a higher-dimensional version of a ( affine ) Toda
field theory, in which a scalar field of the theory takes its value on the
Cartan subalgebra of Kac-Moody algebra. After a diagonal breaking scheme takes
place, such a higher-dimensional version of affine Toda field theory-like
model acquires a quasi-Heisenberg algebra. A case beyond a diagonal breaking
causes more complicated result. An affine Lie algebra can be interpreted as a
special form of trivial fiber bundle, ${\bf g}\otimes{\bf C}[t,t^{-1}]$. Hence
a $G$-bundle and a Maurer-Cartan form of Cartan geometry can, at least
formally, be considered.

Another different perspective is coming from an ${\cal N}=2$ superconformal
algebra, especially the so-called coset construction. It is constructed by the
affine Kac-Moody algebra of $SU(2)$ at level $l$,

$\displaystyle[h_{m},h_{n}]=2ml\delta_{n+m,0},\quad[e_{m},f_{n}]=h_{m+n}+ml\delta_{m+n,0},$
$\displaystyle[h_{m},e_{n}]=2e_{m+n},\quad[H_{m},f_{n}]=-2f_{m+n},$ (67)

with an associated set of complex Grassmann variables. Thus, if a generic
Lagrangian of an NG sector ( for example, see (104) ) is pairwisely decomposed
via taking VEVs, as the direct sum of VEVs of algebra of Lie$(SL(2))$-triple
$(h,e,f)$, then the algebra inside the Lagrangian is a finite and special
version of the $SU(2)$ affine Lie algebra: The algebra arised in the generic
NG-boson Lagrangian can be embedded into the superconformal algebra. This type
of discussion is useful for us to consider several relations between our
generic NG-boson Lagrangian and other theoretical models. It is a known fact
that a class of infinite-dimensional simple linearly compact Lie superalgebras
contain the Standard Model gauge group $SU(3)\otimes SU(2)\otimes U(1)$ as the
algebra of level zero. Our argument presented here has some similarity with
such a situation. The important issue is to know how geometries of these Lie
groups are related with each other. It might be possible to obtain affine Lie
groups starting from Lie algebras of Riemann/Hermitian symmetric spaces.

###  2.4 Graded Lie Algebras and Lie Superalgebras

Lie superalgebras and Lie supergroups have quite interesting characters, and
they have importances in their own right [36,89], while they also acquire
attention from some particle phenomenological point of view. An extension of
our anomalous NG theorem to supersymmetric theory is an interesting subject
for us to complete our theorem. This will be done in another paper by the
author ( in preparation ), and here we will see some perspectives especially
from mathematics. For example, we can consider the following diagram:

Lie superalgebra $\to$ Heisenberg superalgebra ( bosonic/fermionic ) $\to$
Galois supergroup.

The notion of Galois supergroup is not strange, if we consider a non-trivial
central extension to give the supergroup. Via an effective potential and an
order parameter, notions of supermodules, superschemes would be introduced in
our theory of NG theorem. For our context of this paper, the following diagram
is considered:

Lie superalgebras $\to$ Lie supergroups $\to$ supergroup-schemes $\to$
superschemes $\to$ super-étale cohomology $\to$ super-Galois representations,

and,

super-sheaves and supermodules $\to$ superschemes $\to$ perverse super-sheaves
$\to$ super-intersection cohomology $\to$ stratified super-Morse theory,

and,

supergroups $\to$ Maurer-Cartan superforms $\to$ Cartan supergeometry

##  3 The Effective Potential Formalism

Let us examine the effective action $\Gamma_{eff}$ and effective potential
$V_{eff}$ of a general situation. ( See the book of Kugo [49]. ) Let $\Phi$ be
a matrix order parameter. $\Phi$ can be regarded as a left $G$-module [20,30]:
$g(\Phi_{1}+\Phi_{2})=g\Phi_{1}+g\Phi_{2}$, $g\in G$,
$\Phi_{1},\Phi_{2}\in\Phi$. $\Phi$ is assumed as $G$-equivariant. Both $G$ and
$\Phi$ are defined over the same field $F$, usually ${\bf C}$ or ${\bf R}$,
and thus $G$ and $\Phi$ acquire the same topology with $F$. Since $G$ acts on
$\Phi$ continuously, $\Phi$ is a topological $G$-module. A group (co)homology
$H_{n}(G,\Phi)$ and $H^{n}(G,\Phi)$ can be considered by modules generated by
$g\Phi_{1}-\Phi_{1}$ ( difference ), $\Phi_{1}\in\Phi$ and $g\in G$, under the
systematic manner. There are several nontrivial issues in our situation due to
the nature of quantum field theory ( $\Phi$ is a quantum field, not exactly
the same with ${\bf R}$, ${\bf C}$, or ${\bf Z}$ ), as a physical system. If
we regard the effective potential $V_{eff}$ as a scheme, or when $V_{eff}$
defines an algebraic variety, then the (co)homology groups $H_{n}(X,{\cal
O}_{X})$ and $H^{n}(X,{\cal O}_{X})$ ( ${\cal O}_{X}$; a sheaf ) can also be
considered [30,31]. ( Such a cohomology group is introduced anywhere we meet a
sheaf in our theory. The étale cohomology is a cohomology theory of sheaves in
the étale topology [53]. It may be possible to apply the method of étale
cohomology to study a topological nature of our NG theorem. ) If $V_{eff}$ is
included in a line bundle, the Borel-Weil theory can be applied [43].

Let us consider a linear displacement of a field $\Phi$ caused by a conserved
charge $Q^{A}$:

$\displaystyle\delta_{A}\Phi$ $\displaystyle=$
$\displaystyle[Q^{A},\Phi]=\theta^{A}T^{A}\Phi,\quad(A=1,\cdots,N),\quad
T^{A}\in{\rm Lie}(G).$ (68)

A typical example is the chiral $\gamma_{5}$ transformation:
$[Q^{5},\bar{\psi}i\gamma_{5}\psi]=-2\bar{\psi}\psi$, and
$\langle\bar{\psi}\psi\rangle\neq 0$ gives an order parameter with a fixed
phase of the chiral rotation. By takings its VEV, we get

$\displaystyle\langle 0|[Q^{A},\Phi]|0\rangle$ $\displaystyle=$
$\displaystyle\langle 0|\theta^{A}T^{A}\Phi|0\rangle.$ (69)

With taking into account $Q^{A}|0\rangle=0$ ( symmetric ) and
$Q^{A}|0\rangle\neq 0$ ( broken ), usually we conclude that $\langle
0|\theta^{A}T^{A}\Phi|0\rangle=0$ ( symmetric ) and $\langle
0|\theta^{A}T^{A}\Phi|0\rangle\neq 0$ ( broken ). However, in the case of
$SU(2)\to U(1)$ of a ferromagnet, $\langle 0|Q^{z}|0\rangle=\int d^{3}{\bf
x}\langle 0|j^{z}_{0}(x_{0},{\bf x})|0\rangle\neq 0$ may take place even
though $Q^{z}$ is unbroken. Next, we give a formal expansion of
$V_{eff}[\Phi]$ by the set of vectors of ${\rm Lie}(G)$ around the VEV:

$\displaystyle V_{eff}[\Phi]$ $\displaystyle=$ $\displaystyle
V_{eff}[v]+\Bigl{(}\frac{\partial
V_{eff}}{\partial\Phi}\Bigr{)}_{\Phi=v}(\theta^{A}T^{A}\Phi)+\frac{1}{2!}\Bigl{(}\frac{\partial^{2}V_{eff}}{\partial\Phi^{2}}\Bigr{)}_{\Phi=v}(\theta^{A}T^{A}\Phi)^{2}+\cdots.$
(70)

Here $v$ implies a certain type of VEV of $\Phi$. The effective potential
$V_{eff}$ belongs to a germ of a sheaf of smooth function ${\cal O}_{D}$ (
$D$: a domain ). Thus, after taking a VEV of this expansion, the stationary
condition gives the following criterion ( Eqs. (16)-(18) of the paper of
Goldstone, Salam, and Weinberg [27] ):

$\displaystyle\Bigl{(}\frac{\partial^{2}V_{eff}}{\partial\Phi^{2}}\Bigr{)}_{\Phi=v}\langle
0|[Q^{A},\Phi]|0\rangle=\Bigl{(}\frac{\partial^{2}V_{eff}}{\partial\Phi^{2}}\Bigr{)}_{\Phi=v}\langle
0|\theta^{A}T^{A}\Phi|0\rangle=0.$ (71)

This equation is coming from the Nöther theorem of a conserved current
combining with the stationary a condition of $V_{eff}$. Since the zeroth
cohomology group of a Lie algebra is defined by the set of invariants (
annihilated ) under the algebra operation on a module [20],

$\displaystyle H^{0}({\rm Lie}(G),M)$ $\displaystyle=$ $\displaystyle M^{{\rm
Lie}(G)}=\bigl{\\{}m\in M|gm=0,\,\forall g\in{\rm Lie}(G)\bigr{\\}}.$ (72)

Thus the matrix $\frac{\partial^{2}V_{eff}}{\partial\Phi^{2}}$ is interpreted
as an ”effective” invariant module, the zeroth cohomology of the Lie algebra,
via taking VEVs in the quantum theory.
$\frac{\partial^{2}V_{eff}}{\partial\Phi^{2}}$ is regarded to take its value
in the sheaf of germs of continuous functions. ( For example, when $M={\bf
R}$, any connected compact semisimple Lie group $G$ has $H^{0}({\rm
Lie}(G),{\bf R})={\bf R}$, $H^{1}({\rm Lie}(G),{\bf R})=H^{2}({\rm
Lie}(G),{\bf R})=0$. ) In the usual case, the equation (68) has exactly the
same dimension with the number of broken generators and closed in the linear
space, while in the case such as a ferromagnet, $\langle 0|Q^{z}|0\rangle\neq
0$. Note that this VEV will be rewritten after a change of basis set of Lie
algebra, an algebra homomorphism, i.e., $\langle 0|Q^{z}|0\rangle=\langle
0|Q^{x^{\prime}}+Q^{y^{\prime}}+Q^{z^{\prime}}|0\rangle\neq 0$, and thus the
final result obtained from any calculation must not depend on the choice of
Lie algebra representation, $\langle Q^{z}\rangle\neq 0$. This fact means that
a rotation of the frame $(x,y,z)$ must not affect on the physical content of
this equation, of course. The important point is that this equation (68)
cannot be written down only by the set of broken generators in the
ferromagnetic case: This case apparently breaks the condition of proof of the
ordinary NG theorem, and one cannot conclude the existence of a zero-mass
bosonic particle. In other words, the NG boson subspace interacts with the
”symmetric” subspace in a breaking scheme caused by a quantum effect: The
author argues that this is a kind of quantum geometry. Thus, the mass matrix
of NG bosons,

$\displaystyle\Bigl{(}\frac{\partial^{2}V_{eff}}{\partial\Phi^{2}}\Bigr{)}_{\Phi=v}$
$\displaystyle=$ $\displaystyle\Delta^{-1}_{F}(p=0)$ (73)

( $\Delta_{F}(p)$; a matrix Feynman propagator of NG bosons ) ”should” have a
nonzero value. This is just the mechanism of famous Nielsen-Chadha anomaly in
the NG theorem [1,8,34,55,77,81,82,83,84]. Simultaneously, it is also clear
from our discussion, a Lorentz-invariant system with a breaking scheme $G/H$
which gives a symmetric space never has a ”spontaneous violation” of the
ordinary/normal NG theorem. ( The Nielsen-Chadha anomaly never takes place. )
Hence, the ordinary NG theorem is protected by the Coleman-Mandula theorem of
$S$-matrix. Not only the mass matrix of NG bosons, but the dispersion
relations themselves should be derived from
$\frac{\partial^{2}V_{eff}}{\partial\Phi^{2}}$. It should be mentioned that
there might be a similar situation in a relativistic model with a Lorentz-
violating parameter: For example, the following VEV in an $SU(2)$ isospin
space could be considered,

$\displaystyle\langle\bar{\psi}\tau_{3}\psi\rangle\neq 0,$ (74)

in a NJL type model. In this VEV, $\tau_{3}$ is symmetric while
$(\tau_{1},\tau_{2})$ is broken. It might give a similar situation with the
ferromagnet with $\langle Q^{3}_{isospin}\rangle\neq 0$.

Let us discuss further on $V_{eff}$. From the general theory of effective
action, the displacement (65) derives the following equation, from a second-
order derivative of the effective potential $V_{eff}$ after taking a VEV:

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\langle(V^{\prime\prime}_{eff})_{1}\theta^{1}T^{1}\Phi+\cdots+(V^{\prime\prime}_{eff})_{N}\theta^{N}T^{N}\Phi\rangle$
(75) $\displaystyle=$
$\displaystyle\langle(V^{\prime\prime}_{eff})_{1}\theta^{1}T^{1}v+\cdots+(V^{\prime\prime}_{eff})_{N}\theta^{N}T^{N}v\rangle,$
$\displaystyle V^{\prime\prime}_{eff}$ $\displaystyle=$
$\displaystyle\frac{\delta^{2}V_{eff}[\Phi]}{\delta\Phi^{2}},$ (76)
$\displaystyle\Phi$ $\displaystyle=$ $\displaystyle
c_{1}T^{1}+\cdots+c_{N}T^{N},\quad\\{c_{j}\\}\in{\bf C},\,(j=1,\cdots,N).$
(77)

Here $v\in M_{n}({\bf C})$ ( matrix ) indicates a VEV. $\Phi$ is a $G$-module
expanded by the basis of ${\rm Lie}(G)$. Some normalization condition for
$\Phi$ is set aside for a while. Since $T^{A}\Phi$ is cased by taking
adjoints, the above equation implicitly contains the root space of the Lie
algebra ( i.e., from $[{\bf h},{\bf g}]=\lambda({\bf h}){\bf g}$,
$\lambda({\bf h})$; root, ${\bf h}$; Cartan subalgebra, ${\bf g}\in{\rm
Lie}(G)$ ), and the corresponding Weyl group acts implicitly. In a case of
Riemannian symmetric space, the adjoints, the Killing form, and the Jacobi
field are obtained from its root system ( see the book of Helgason [32] ).
Then they are related with a harmonic mapping and a harmonic analysis. Later,
we will mention that an NG boson gives a geodesic in a case of Riemannian
symmetric space. Note that the ${\rm Lie}(G)$ itself can be regarded as a
$G$-module, by satisfying the axiom of a $G$-module with a certain type of
group operations $G\times{\rm Lie}(G)\to{\rm Lie}(G)$. This formula (72)
includes the ”off-diagonal” contributions of the second-order derivative of
$V_{eff}$ in the space of Lie algebra generators which may cause mode-mode
couplings of NG bosons: Later, we will observe that a mode-mode coupling
between bosonic fields modifies dispersion relations of NG bosons and as a
consequence, an NG boson acquires a finite mass. Needless to say, the bases
$T^{A}\in{\rm Lie}(G)$ are always linearly independent. For example, in a case
of $SU(2)\to U(1)$,

$\displaystyle 0$ $\displaystyle=$
$\displaystyle(V^{\prime\prime}_{eff})_{1}\langle[Q^{1},\Phi]\rangle+(V^{\prime\prime}_{eff})_{2}\langle[Q^{2},\Phi]\rangle+(V^{\prime\prime}_{eff})_{3}\langle[Q^{3},\Phi]\rangle,$
(78) $\displaystyle\Phi$ $\displaystyle=$ $\displaystyle
c_{1}\sigma^{1}+c_{2}\sigma^{2}+c_{3}\sigma^{3}.$ (79)

Here, $\Phi\in{\bf 2}$-representation. We write the Lie brackets in the above
equation as follows:

$\displaystyle\langle[Q^{1},c_{2}\sigma^{2}]\rangle=(c_{2})_{1}\sigma_{3},\quad\langle[Q^{1},c_{3}\sigma^{3}]\rangle=(c_{3})_{1}\sigma_{2},$
$\displaystyle\langle[Q^{2},c_{1}\sigma^{1}]\rangle=(c_{1})_{2}\sigma_{3},\quad\langle[Q^{2},c_{3}\sigma^{3}]\rangle=(c_{3})_{2}\sigma_{1},$
$\displaystyle\langle[Q^{3},c_{1}\sigma^{1}]\rangle=(c_{1})_{3}\sigma_{2},\quad\langle[Q^{3},c_{2}\sigma^{2}]\rangle=(c_{2})_{3}\sigma_{1}.$
(80)

Then we get

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\bigl{\\{}(V^{\prime\prime}_{eff})_{1}(c_{2})_{1}+(V^{\prime\prime}_{eff})_{2}(c_{1})_{2}\bigr{\\}}\sigma_{3}$
(81)
$\displaystyle+\bigl{\\{}(V^{\prime\prime}_{eff})_{2}(c_{3})_{2}+(V^{\prime\prime}_{eff})_{3}(c_{2})_{3}\bigr{\\}}\sigma_{1}$
$\displaystyle+\bigl{\\{}(V^{\prime\prime}_{eff})_{1}(c_{3})_{1}+(V^{\prime\prime}_{eff})_{3}(c_{1})_{3}\bigr{\\}}\sigma_{2}$

Due to the linear independence of $\sigma_{1,2,3}$, all of the coefficients
vanish. If $\langle[Q^{1},\Phi]\rangle\neq 0$ and/or
$\langle[Q^{2},\Phi]\rangle\neq 0$, both of them have contributions to rotate
$\Phi$ to the third direction proportional to $\sigma^{3}$, and there is a
freedom to take $(V^{\prime\prime}_{eff})_{1,2,3}$ finite in those vanishing
coefficients ( the case of our anomalous NG theorem of a ferromagnet ), while
if $\langle[Q^{1},\Phi]\rangle=\langle[Q^{2},\Phi]\rangle=0$, then
$(V^{\prime\prime}_{eff})_{1,2,3}$ will vanish independently ( the case of
ordinary NG theorem ). More general case is understood by the algebra we have
discussed in the previous section: At least in the case of quasi-Heisenberg
algebra, a mode-mode coupling takes place in $V_{eff}$ which becomes apparent
from its second-order derivatives. Hence, we arrive at the following theorem:

Theorem: Any type of mode-mode coupling between NG bosons modifies their
dispersion relations and mass spectra, gives an anomalous behavior of the NG
theorem.

We also yield another simple but important result:

Theorem: Any spontaneous symmetry breaking of isolated $U(1)$ Abelian Lie
group cannot give an anomalous behavior of the NG theorem, due to the lack of
mode-mode coupling.

In the theory of itinerant (anti)ferromagnetism of Moriya [56,57], he pointed
out that a mode-mode coupling of magnetic ( i.e., spin ) fluctuations is
important which can be experimentally observed by the method of magnetic
resonance. Our anomalous NG theorem of a ferromagnet may have a physical
implication in the Moriya theory. If the mass of massive NG boson is
controlled as a function of a strength external field or temperature, the
energy split of two NG bosons might be made small relative to the
characteristic energy scale. In the vicinity of the critical region, those
bosons acquire special importance. In the theory of Moriya, the method of
self-consistent renormalization theory is employed, which takes into account
diagrammatically higher-order interactions between electrons. It is
interesting from our context that how such higher-order interactions affect
the mass spectra of NG bosons, simultaneously to the value of $T_{c}$ and
correlation lengths.

In a composite particle model, a Schwinger-Dyson equation determines an order
parameter which is non-local, for example a two-point function $\langle
T\phi(x)\phi(y)\rangle\neq 0$ [15]. In that case, the gap equation is derived
by ( in a translation-invariant case )

$\displaystyle\frac{\delta V_{eff}(G(p))}{\delta G(p)}$ $\displaystyle=$
$\displaystyle 0.$ (82)

Here, $G(p)$ is a propagator. Now, the mass matrix of NG bosons may be
determined by the examination of the following equation:

$\displaystyle\sum^{N}_{A=1}\Bigl{(}\frac{\delta^{2}V_{eff}(G(p))}{\delta
G(p)^{2}}\Bigr{)}_{A}T^{A}G(p)$ $\displaystyle=$ $\displaystyle 0.$ (83)

Here, $T^{A}G(p)$ imply VEVs, for example,
$\langle\bar{\psi}T^{A}\psi\rangle$. Now, the Lie group $G$ acts on the
propagator $G(p)$ nontrivially, though the algebra we consider here is an
adjoint type, and thus this equation is essentially the same with (68) as an
equation of Lie algebra. Hence we obtain a mathematically similar equation
with (72). We conclude that the essential part of the basis of our anomalous
NG theorem is the same in case of both composite and elementary fields.

##  4 The Kaon Condensation Model

Let us start from the following $SU(2)$ Higgs-Kibble-type model Lagrangian:

$\displaystyle{\cal L}$ $\displaystyle=$ $\displaystyle-\frac{1}{4}{\rm
tr}F^{a}_{\mu\nu}F^{a\mu\nu}+\tilde{D}^{\dagger}_{\nu}\Phi^{\dagger}\tilde{D}^{\nu}\Phi+m^{2}_{0}\Phi^{\dagger}\Phi-\frac{\lambda}{2}(\Phi^{\dagger}\Phi)^{2}+\epsilon
m^{2}_{ex}\Phi^{\dagger}\sigma^{3}\Phi,$ $\displaystyle\tilde{D}_{\nu}$
$\displaystyle=$
$\displaystyle\partial_{\nu}-i\mu\delta_{0\nu}-\frac{i}{2}g\sigma^{a}A^{a}_{\nu},$
(84)

Here, we introduce $SU(2)$-gauge fields $A^{a}_{\nu}$ ( $a=1,2,3$ ), and the
mass term $\epsilon m^{2}_{ex}\Phi^{\dagger}\sigma^{3}\Phi$ with a relatively
small parameter $\epsilon$ explicitly breaks the symmetry. Since this explicit
symmetry breaking parameter breaks the symmetries of $\sigma^{1}$ and
$\sigma^{2}$, both of them acquire finite masses. The complex bosonic field is
defined as $\Phi\equiv(\phi_{1},\phi_{2})^{T}$ as a ${\bf 2}$-representation,
and $\mu$ is a Lorentz-symmetry violating chemical potential. Let us mention
the fact that the anomalous behavior of NG bosons cannot be understandable by
the hypercharge model of $U(2)$ used in the paper of Schaefer et al [77]: Due
to the special nature of $SU(2)$, this model has an additional symmetry, so-
called ”custodial symmetry” of $SU(2)$. Hence the breaking scheme of this
model is $SU(2)\otimes SU(2)\simeq SO(4)\to SU(2)_{diag}$. This breaking
scheme is essentially the same with the Lie algebra of $SU(N)_{L}\otimes
SU(N)_{R}\to SU(N)_{V}$ discussed at (61). Namely,
$[\theta^{a}\sigma^{a}\otimes
1+\varphi^{b}\sigma^{b}\otimes\sigma^{3},\sum^{3}_{j=1}a_{j}\sigma^{j}]$ will
be examined. We consider the broken and symmetric generators carefully
according to this breaking scheme. First, let us consider the case where all
of the gauge fields are dropped from this model. We assume the model chooses
$\Phi_{0}=\langle\Phi\rangle=(0,v)^{T}/\sqrt{2}$ as one of its vacua: Then we
yield the following relation for the VEV $v$:

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\frac{\partial
V^{(tree)}_{eff}}{\partial v},$ (85) $\displaystyle V^{(tree)}_{eff}$
$\displaystyle=$ $\displaystyle-\frac{1}{2}(\mu^{2}+m^{2}_{0}-\epsilon
m^{2}_{ex})v^{2}+\frac{\lambda}{8}v^{4},$ (86) $\displaystyle v$
$\displaystyle=$ $\displaystyle\pm\sqrt{\frac{2(\mu^{2}+m^{2}_{0}-\epsilon
m^{2}_{ex})}{\lambda}}.$ (87)

Now a small displacement around the vacuum solution we have obtained, consists
with the amplitude mode and three NG modes, is described by the following ’t
Hooft parametrization:

$\Phi=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}\chi_{2}+i\chi_{1}\\\
v+\psi-i\chi_{3}\end{array}\right)$ (88)

Then we get

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Bigg{[}\tilde{\partial}^{\dagger}_{\nu}\psi\tilde{\partial}_{\nu}\psi+\tilde{\partial}^{\dagger}_{\nu}\chi_{a}\tilde{\partial}_{\nu}\chi_{a}+\mu^{2}v^{2}$
(89)
$\displaystyle\quad+i\Bigl{(}\tilde{\partial}^{\dagger}_{\nu}\chi_{2}\tilde{\partial}_{\nu}\chi_{1}-\tilde{\partial}^{\dagger}_{\nu}\chi_{1}\tilde{\partial}_{\nu}\chi_{2}\Bigr{)}+i\Bigl{(}\tilde{\partial}^{\dagger}_{\nu}\chi_{3}\tilde{\partial}_{\nu}v-\tilde{\partial}^{\dagger}_{\nu}v\tilde{\partial}_{\nu}\chi_{3}\Bigr{)}$
$\displaystyle\quad-i\Bigl{(}\tilde{\partial}^{\dagger}_{\nu}\psi\tilde{\partial}_{\nu}\chi_{3}-\tilde{\partial}^{\dagger}_{\nu}\chi_{3}\tilde{\partial}_{\nu}\psi\Bigr{)}+\tilde{\partial}^{\dagger}_{\nu}\psi\tilde{\partial}_{\nu}v+\tilde{\partial}^{\dagger}_{\nu}v\tilde{\partial}_{\nu}\psi\Bigg{]}$
$\displaystyle\quad+\frac{m^{2}_{0}}{2}\bigl{(}\chi^{2}_{1}+\chi^{2}_{2}+\chi^{2}_{3}+(v+\psi)^{2}\bigr{)}$
$\displaystyle\quad-\frac{\lambda}{8}\bigl{(}\chi^{2}_{1}+\chi^{2}_{2}+\chi^{2}_{3}+(v+\psi)^{2}\bigr{)}^{2}$
$\displaystyle\quad+\frac{\epsilon
m^{2}_{ex}}{2}\bigl{(}\chi^{2}_{1}+\chi^{2}_{2}-\chi^{2}_{3}-(v+\psi)^{2}\bigr{)},$
$\displaystyle\tilde{\partial}_{\nu}$ $\displaystyle=$
$\displaystyle\partial_{\nu}-i\mu\delta_{0\nu}.$ (90)

After using the expression of the VEV $v$ ( Eq.(84) ) of $\Phi$, we get the
quadratic part of the Lagrangian in the following form:

$\displaystyle{\cal L}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bigl{(}\partial_{\nu}\psi\partial^{\nu}\psi+\partial_{\nu}\chi_{a}\partial^{\nu}\chi_{a}\bigr{)}$
(91)
$\displaystyle+\mu\bigl{(}\chi_{1}\partial_{0}\chi_{2}-\chi_{2}\partial_{0}\chi_{1}+\chi_{3}\partial_{0}\psi-\psi\partial_{0}\chi_{3}\bigr{)}$
$\displaystyle+\epsilon
m^{2}_{ex}(\chi^{2}_{1}+\chi^{2}_{2})-(\mu^{2}+m^{2}_{0}-\epsilon
m^{2}_{ex})\psi^{2}.$

$\epsilon>0$ must be excluded to avoid tachyonic modes in
$(\chi_{1},\chi_{2})$, and this condition holds for the dispersion relations
after diagonalization of ${\cal L}^{(2)}$ ( see, (92) ) Note that the total
Lagrangian ${\cal L}$ is a class function ${\cal L}(\Phi)={\cal L}(g\Phi
g^{-1})$, $g\in G$, while both the effective potential
$V_{eff}(\Phi=\Phi_{0})$ and ${\cal L}^{(2)}(\delta\Phi=\Phi-\Phi_{0})$ are
not class functions, even if the explicit symmetry breaking parameter
vanishes. This fact is crucially important for us to understand the physics of
our anomalous NG theorem. In the case of normal NG theorem,
$V_{eff}(\Phi=\Phi_{0})$ is ”effectively” a constant under the action of $g\in
G$ since it does not contain any coordinate of the NG manifold explicitly.
While, in our anomalous NG theorem, $V_{eff}(\Phi=\Phi_{0})$ will acquire an
energy=mass by the action of $g\in G$, and the energy which corresponds to the
mass of an NG boson is provided from ${\cal L}^{(2)}(\delta\Phi)$. Therefore,
the effective potential of the theory is certainly periodically modulated
under the action of $g\in G$ if $G$ is compact. We will see this is the case
in our discussion given below. By using the path-integral formalism $\int{\cal
D}\Phi{\cal D}\Phi^{\dagger}e^{i\int{\cal L}(\Phi,\Phi^{\dagger})}$, we
immediately recognize that this phenomenon of anomalous NG theorem is a pure
quantum effect, since the Lagrangian of the beginning, ${\cal
L}(\Phi,\Phi^{\dagger})$, is also a class function of any group operation
$g\in G$ when any explicit symmetry breaking parameter vanishes. Namely, it is
impossible to understand it by a tree-level, and we need at least the one-loop
level to obtain the finite curvature along with a massive NG-bosonic
coordinate. This is a remarkable fact since the treatment of the symmetry
breaking of the traditional bosonic Goldstone model ( and also the Standard
Model Higgs sector ) is understandable at the tree level, while a fermion
composite model such as an NJL-type model has to evaluate at least a one-loop
effective potential. To make the Lagrangian in a Hermitian matrix form
explicitly, we perform partial integrations in time-derivatives and rearrange
the Lagrangian:

${\cal
L}^{(2)}=\frac{1}{2}\tilde{\Phi}\left(\begin{array}[]{cccc}k^{2}+\epsilon
m^{2}_{ex}&ik_{0}\mu&0&0\\\ -ik_{0}\mu&k^{2}+\epsilon m^{2}_{ex}&0&0\\\
0&0&k^{2}&ik_{0}\mu\\\
0&0&-ik_{0}\mu&k^{2}-M^{2}\end{array}\right)\tilde{\Phi},$ (92)

where,

$\displaystyle\tilde{\Phi}$ $\displaystyle=$
$\displaystyle(\chi_{1},\chi_{2},\chi_{3},\psi)^{T},$ (93) $\displaystyle
M^{2}$ $\displaystyle=$ $\displaystyle\mu^{2}+m^{2}_{0}-\epsilon m^{2}_{ex}.$
(94)

Then we get

$\displaystyle E^{\chi_{1},\chi_{2}}_{\pm}$ $\displaystyle=$
$\displaystyle\sqrt{{\bf k}^{2}-\epsilon
m^{2}_{ex}+\frac{\mu^{2}}{2}\pm\frac{\mu}{2}\sqrt{\mu^{2}+4{\bf
k}^{2}-4\epsilon m^{2}_{ex}}},$ (95) $\displaystyle E^{\chi_{3},\psi}_{\pm}$
$\displaystyle=$ $\displaystyle\sqrt{{\bf
k}^{2}+\frac{M^{2}+\mu^{2}}{2}\pm\frac{1}{2}\sqrt{4\mu^{2}{\bf
k}^{2}+(M^{2}+\mu^{2})^{2}}}.$ (96)

Now it is clear for us from these dispersion relations at the limit ${\bf
k}\to 0$ and $\epsilon\to 0$, $E^{\chi_{1},\chi_{2}}_{+}$ and
$E^{\chi_{3},\psi}_{+}$ are massive while $E^{\chi_{1},\chi_{2}}_{-}$ and
$E^{\chi_{3},\psi}_{-}$ are massless.

Therefore, we find that the masses of NG bosons in the $SU(2)$-Higgs-Kibble-
type model are coming from the mode-mode coupling of a pair of broken
generators. This fact provides us a confirmation on our general discussion
given in the previous section. Hence, $\langle[Q^{1},Q^{2}]\rangle\neq 0$ and
other commutators must have vanishing VEVs in this case. This is achieved by
the form of the vacuum $\Phi_{0}=(0,v)/\sqrt{2}$. It must be distinguished
that the commutators $[Q^{A},Q^{B}]$ are given by conserved charges and now
$\langle Q^{1}\rangle=\langle Q^{2}\rangle=0$, $\langle Q^{3}\rangle\neq 0$,
while an order parameter can take the form $\Phi\sim
a_{1}\sigma^{1}+a_{2}\sigma^{2}+a_{3}\sigma^{3}$, $a_{1}\neq 0$, $a_{2}\neq
0$, $a_{3}\neq 0$. A Heisenberg algebra is obtained, and a pairwise decoupling
takes place. In this example of kaon condensation, the amplitude mode $\psi$
is defined toward the direction of VEV, and it couples with $\chi_{3}$. This
amplitude mode might be expressed more generally such as
$\psi\to\exp\psi((1-\sigma^{3})/2)\Phi_{0}$, though this is not an element of
the Lie group $SU(2)$, rather a projection operator, and then the manner of
mode-mode coupling between $\psi$ and $\chi_{3}$ is not given in the same
manner of commutator $[Q^{1},Q^{2}]$, different from the mode-mode coupling of
$\chi_{1}$ and $\chi_{2}$. It should be noticed that
$\chi_{1}=\Re\delta\phi_{1}$, $\chi_{2}=\Im\delta\phi_{1}$,
$\psi=\Re\delta\phi_{2}$, $\chi_{3}=\Im\delta\phi_{2}$, where
$(\delta\phi_{1},\delta\phi_{2})$ are fluctuations in the vicinity of the
stationary point. Thus, the mode-mode couplings relevant for our anomalous NG
theorem take place between the real and imaginary parts of $\phi_{1}$ and
$\phi_{2}$ separately. This fact in our kaon condensation model is quite
interesting toward a classification of several possible types of mode-mode
couplings in NG bosons. In the above example, the mode-mode couplings of the
NG bosons and the amplitude are described over a two independent discs of
Gaussian plane. It may be possible to generate a mode-mode coupling which
gives a finite mass to an NG boson via a radiative correction: This can be
regarded as a kind of Coleman-Weinberg mechanism in an NG boson mass matrix.
This can be understandable if there is an interaction between two NG bosons (
such as an electromagnetic interaction ): Especially, a Rayleigh-Schrödinger
or a quasi-degenerate perturbation theory can apply to the case where the
spectrum of an NG sector is split by small but finite masses.

The periodicity of the NG sector inside the Lagrangian can be understood as
follows. Since the massive NG bosons arises from the pair
$(\chi^{1},\chi^{2})$, we prepare

$g=e^{i(\chi_{1}\sigma^{1}+\chi_{2}\sigma^{2})}=\left(\begin{array}[]{cc}\cos|\chi|&i\frac{\chi_{-}}{|\chi|}\sin|\chi|\\\
i\frac{\chi_{+}}{|\chi|}\sin|\chi|&\cos|\chi|\end{array}\right),$ (97)

where,

$\displaystyle|\chi|$ $\displaystyle=$
$\displaystyle\sqrt{\chi^{2}_{1}+\chi^{2}_{2}},\quad\chi_{\pm}=\chi_{1}\pm
i\chi_{2}.$ (98)

Then we evaluate

$\displaystyle\tilde{\partial}_{\nu}g\Phi_{0}$ $\displaystyle=$ $\displaystyle
i\Bigl{[}(\tilde{\partial}_{\nu}\chi_{1})\sigma^{1}+(\tilde{\partial}_{\nu}\chi_{2})\sigma^{2}\Bigr{]}g\Phi_{0},$
(99)

and take the following inner product, we get

$\displaystyle\tilde{\partial}^{\dagger}_{\nu}(\Phi_{0}g^{-1})\cdot\tilde{\partial}_{\nu}(g\Phi_{0})$
$\displaystyle\quad=v^{2}\Bigl{[}\tilde{\partial}^{\dagger}_{\nu}\chi_{1}\tilde{\partial}_{\nu}\chi_{1}+\tilde{\partial}^{\dagger}_{\nu}\chi_{2}\tilde{\partial}_{\nu}\chi_{2}+i\bigl{(}\tilde{\partial}^{\dagger}_{\nu}\chi_{1}\tilde{\partial}_{\nu}\chi_{2}-\tilde{\partial}^{\dagger}_{\nu}\chi_{2}\tilde{\partial}_{\nu}\chi_{1}\bigr{)}\cos
2|\chi|\Bigr{]}.$ (100)

Therefore, the chemical potential $\mu$ acquires a periodic modulation
proportional to the trigonometric function such that $\sim\mu^{2}\cos
2|\chi|$, and thus the mass of the dispersion $E^{\chi_{1},\chi_{2}}_{+}$
becomes periodic as a function of $|\chi|$. Namely, the periodic modulation of
the effective potential is kinematically generated in our anomalous NG theorem
by the mode-mode-coupling caused by a finite chemical potential: This is a
quite remarkable result, since our result explains both the mechanism of
generation of a finite mass of an NG bosons, and a kinematically generated
periodicity of the effective potential beyond the tree-level defined over the
$SU(2)$ group manifold. It should be mentioned that this expression of the
kinetic part of the Lagrangian is obtained by choosing a specific form of the
VEV of $\Phi$, i.e., $\langle\Phi\rangle=\Phi_{0}=(0,v)^{T}/\sqrt{2}$, and
thus this expression is a ”function” of the form of VEV, the special form of
local coordinates $|\chi|$, and the chemical potential $\mu$. Hence, if we
choose another type of VEV to $\Phi$, then in fact we will obtain another
expression different from (97): From this sense, both $V^{(tree)}_{eff}$ and
${\cal L}^{(2)}$ are not class functions of $SU(2)$ ( caution: the kinetic
term given above contains all of the orders of fluctuations
$(\chi_{1},\chi_{2})$, not only ${\cal L}^{(2)}$ ). The form $|\chi|$ implies
that the theory is isotropic toward the directions $\chi_{1}$ and $\chi_{2}$ (
axial symmetric ), similar to the case of a ferromagnet. One should notice
that the Lagrangian of (88) or (89) is defined locally, at a specific point
over the $SU(2)$ Lie group manifold. To see the periodicity, we need a group
element which is defined globally, as we have used above. Now we obtain the
effective potential of the model as $V_{eff}\sim
V^{(tree)}_{eff}+f(v)\mu^{2}\cos 2|\chi|$ by using the result of
$E^{\chi_{1},\chi_{2}}_{+}$ ( $f(v)$: a scalar function of the VEV $v$ ),
which shows a periodicity toward the direction of ”amplitude”
$|\chi|=\sqrt{\chi^{2}_{1}+\chi^{2}_{2}}$, while it is flat to the direction
of the phase ( precession mode ) of $\chi_{1}+i\chi_{2}$ ( the amplitude and
the phase defines an infinite number of $S^{1}$ circles of a Gaussian plane ),
as we have stated above. This fact is parallel with the case of ferromagnet,
where the massless NG mode ( spin wave ) is the precession described by a
linear combination of the two modes $(\sigma_{1},\sigma_{2})$. Absolutely
interesting fact we have found here is that this global structure of the
effective potential ( periodic toward the radial direction $|\chi|$ while
exactly flat along with the phase variable ) is coming from the uncertainty
relation arises from the Heisenberg algebra obtained from $SU(2)$: There is a
strong uncertainly toward the phase direction, while the motion toward the
radial direction is well localized and the ”position” is determined by the set
of periodic stationary points: The set of stationary points gives a Galois
symmetry. In the vicinity of a stationary point ( valley ), a representation
point has a small fluctuation toward the radial direction while it strongly
fluctuate along with the phase coordinate. Namely,

Theorem: The uncertainty relation of the Heisenberg algebra obtained from the
Lie algebra of $SU(2)$ realizes in the global structure of the effective
potential of a theory. One degree of freedom is almost fixed/determined while
another degree of freedom of the Heisenberg pair of shows a strong
uncertainty.

We argue that a quasi-Heisenberg algebra generically obtained in various
symmetry breaking schema of our anomalous NG theorem will determine the
structure of effective potential according to satisfy the uncertainty
relations. Therefore, a mass generation in an NG sector in our anomalous NG
theorem reflects the uncertainty principle! Furthermore, an explicit symmetry
breaking mass parameter such as the prescription of explicitly+dynamical
symmetry breakings also acts to fix a phase degree of freedom, and thus, a
mass spectrum any meson ( mesonic state ) in a non-Abelian Lie group symmetry
is generically a result of uncertainty relation.

The kaon condensation model we consider here has a lot of similar aspects with
physics of a ferromagnet. Needless to say, the time-reversal symmetry is
broken in a ferromagnet, and a precession of magnetization reflects this time-
reversal symmetry breaking. The system of kaon condensation we discuss here
might have an effect of time-reversal symmetry breaking, caused by the NG
bosons, at its low-energy excited state.

We make a brief comment on the Higgs ( Anderson-Higgs-Brout-Englert-Guralnik-
Hagen-Kibble ) phenomenon in the Lagrangian (81) [2,21,28,34]. If we put a
$U(1)$ gauge field to the Lagrangian, it also gives a Higgs phenomenon with
the field redefinition $U_{0}=\mu+A_{0}-\partial_{0}\theta$,
$U_{i}=A_{i}-\partial_{i}\theta$ ( $\theta$: a $U(1)$ phase ) and thus they
give a massive Proca theory. While if we put a set of $SU(2)$ gauge fields as
(81), the chemical potential $\mu$ plays no role and the usual Higgs
phenomenon is observed.

##  5 The Model Lagrangian Approach

In all of the above classification of types of NG bosons given in the end of
the introduction of this paper, the dispersion relations and mass spectrum of
NG bosons should be obtained from an analysis of effective action $V_{eff}$.
In general, an effective action and/or a potential are expanded by bosonic
fields, thus we restrict ourselves to the case of bosonic ( bosonized )
theory. Since the NG boson Lagrangian will be obtained from $V_{eff}$, we
construct a Lagrangian to make our problem more tractable. Let ${\cal
L}(\Phi)$ be a Lagrangian, and let us take a small displacement of bosonic
field as $\Phi=\Phi_{0}+\delta\Phi$. $\Phi$ is assumed to belong to a
representation of $G$. $\delta\Phi$ contains the NG bosons and the amplitude
mode. In the case of $SU(N)$, one frequently use a fundamental representation,
${\bf N}\ni\Phi,\Phi_{0},\delta\Phi$. Let us consider the case of symmetry
given by a Lie group $G$ with ${\rm dim}{\rm Lie}(G)=N$. Then ${\rm
dim}(\delta\Phi)=N+1$, where the additional one degree of freedom is the
amplitude mode of $\Phi$. Then the Lagrangian is expanded into the following
form:

$\displaystyle{\cal L}(\Phi_{0}+\delta\Phi)$ $\displaystyle=$
$\displaystyle{\cal L}(\Phi_{0})+\frac{\partial{\cal
L}(\Phi_{0})}{\partial(\delta\Phi)}\delta\Phi+\frac{1}{2!}\frac{\partial^{2}{\cal
L}(\Phi_{0})}{\partial(\delta\Phi)^{2}}(\delta\Phi)^{2}+\cdots.$ (101)

The first-order derivative vanishes in the effective action, and the second-
order derivative gives the mass matrix and dispersion relations of NG bosons.
Namely,

$\displaystyle\frac{\partial^{2}{\cal L}(\Phi_{0})}{\partial(\delta\Phi)^{2}}$
$\displaystyle=$ $\displaystyle\Delta^{-1}_{F}(p_{\nu}).$ (102)

This equation corresponds to (70). The algebraic roots of ${\rm
det}\Delta^{-1}_{F}(p_{\nu})=0$ gives the dispersion relations of NG bosons.

One can also consider the case where the order parameter is a vector/tensor
$\Phi_{\mu\nu\cdots\rho}$, explicitly breaks the Lorentz symmetry. In that
case, we can consider the following formal expansion:

$\displaystyle{\cal L}(\Phi_{\mu\nu\cdots\rho})$ $\displaystyle=$
$\displaystyle\sum^{\infty}_{n=0}\frac{1}{n!}\frac{\partial^{n}{\cal
L}}{\partial(\delta\Phi_{\mu\nu\cdots\rho})^{n}}(\delta\Phi_{\mu\nu\cdots\rho})^{n}.$
(103)

Here, we do not consider contractions of the Lorentz indices
$(\mu,\nu,\cdots,\rho)$. A vectorial order parameter is frequently found in
superconductivity or 3He superfluidity [50,65,66,78]. The second-order
derivative term as the quadratic part of quantum fluctuations,

$\displaystyle{\cal L}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{\partial^{2}{\cal
L}}{\partial(\delta\Phi_{\mu\nu\cdots\rho})^{2}}(\delta\Phi_{\mu\nu\cdots\rho})^{2}$
(104)

gives the mass matrix and dispersion relations of NG bosons. While, due to the
Coleman-Mandula theorem, a Poincaré-invariant theory can have only a conserved
charge of scalar type. We currently consider a Lorentz-violating system of
(non)relativistic theory, and thus we assume a theory can have vector/tensor-
charges $Q^{A}_{\mu\nu\cdots\rho}$ of internal symmetries. Hence, formally,

$\displaystyle\delta^{A}_{\mu\nu\cdots\rho}\Phi^{A^{\prime}}_{\mu^{\prime}\nu^{\prime}\cdots\rho^{\prime}}$
$\displaystyle=$
$\displaystyle[Q^{A}_{\mu\nu\cdots\rho},\Phi^{A^{\prime}}_{\mu^{\prime}\nu^{\prime}\cdots\rho^{\prime}}]$
(105)

will be considered. Hence, we find that the Lie brackets
$[Q^{A}_{\mu\nu\cdots\rho},\Phi^{A^{\prime}}_{\mu^{\prime}\nu^{\prime}\cdots\rho^{\prime}}]$
define how ${\cal L}^{(2)}$ is given in terms of the NG bosons, similar to the
case of $V_{eff}$ we have discussed in the previous section. Namely, we will
consider the Lie brackets of internal symmetries with Lorentz indices. Thus, a
quasi-Heisenberg algebra should be obtained from
$[Q^{A}_{\mu\nu\cdots\rho},Q^{B}_{\mu^{\prime}\nu^{\prime}\cdots\rho^{\prime}}]$
in a diagonal symmetry breaking scheme. If this algebra causes a mode-mode
coupling in quantum fluctuations of NG bosons, then our anomalous NG theorem
takes place. Hence, we argue it is enough for us to consider a scalar field to
study the mechanism of our anomalous NG theorem.

Now, we can systematically construct a generic Lagrangian which may show the
phenomenon of anomalous NG theorem. We know from our observation on the model
of kaon condensation, the relevant part of the Lagrangian to give the
anomalous behavior of NG bosons is its kinetic term. For example, in the case
of two-component real bosonic field:

${\cal
L}=\frac{1}{2}(\phi_{1},\phi_{2})\left(\begin{array}[]{cc}\partial_{\nu}\partial^{\nu}&a_{\nu}\partial^{\nu}\\\
a^{*}_{\nu}\partial^{\nu}&\partial_{\nu}\partial^{\nu}\end{array}\right)\left(\begin{array}[]{c}\phi_{1}\\\
\phi_{2}\end{array}\right).$ (106)

The off-diagonal part describes a mode-mode coupling: From our examination on
the Lagrangian of NG sector of the kaon condensation model, it is now obvious
fact that the explicit symmetry breaking mass parameter enters into the
diagonal part of the Lagrangian matrix, while the mode-mode coupling matrix
elements take of course several off-diagonal elements. We do not consider
kinetic terms of derivatives higher than the second-order, since they may
cause a tachyonic mode or a negative norm state. Therefore, this mechanism of
anomalous NG theorem must break the Lorentz symmetry.

With respect to the general theory of nonlinear sigma models, the following
Lagrangian is examined:

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\delta\Phi{\cal M}\delta\Phi$ (110) $\displaystyle=$
$\displaystyle\frac{1}{2}g_{ab}\partial_{\nu}(\Psi)^{a}\partial^{\nu}(\Psi)^{b}-\frac{1}{2}\Psi
M^{2}\Psi+\Psi\left(\begin{array}[]{ccc}0&&\sum\tilde{a}_{\nu}\partial^{\nu}\\\
&\ddots&\\\
\sum\tilde{a}^{\dagger}_{\nu}\partial^{\nu}&&0\end{array}\right)\Psi,$
$\displaystyle\Psi$ $\displaystyle=$
$\displaystyle(\chi_{1},\cdots,\chi_{N},\phi)^{T},\quad(a=1,2,\cdots,N).$
(111)

Here, the real matrix $g_{ab}$, which defines a Riemannian metric
$\hat{g}=g_{ab}dx^{a}\otimes dx^{b}$ ( $dx^{a},dx^{b}\in{\bf R}$), must
satisfy a condition given by the Lie group $G$. Usually, it is taken to be a
unit matrix, i.e., which defines a Euclidean space, a typical example of a
simply connected Riemannian symmetric space, with its dimension equal to the
number of broken generators. It is known fact that, in a small displacement
$\Psi\to\Psi+\delta\Psi$, $\delta\Psi$ must be a Killing vector defined over
the target space of a nonlinear sigma model. Here, we have put explicit
symmetry breaking mass parameters $M^{2}$ ( a Hermitian matrix ).
$\tilde{a}_{\nu}$ are matrices, and they can contain both explicit symmetry
breaking parameters ( like a chemical potential ) or spontaneously generated
parameters from its underlying theory. Therefore, the number of roots of

$\displaystyle{\rm det}{\cal M}(p_{\nu}\neq 0)$ $\displaystyle=$
$\displaystyle 0$ (112)

gives the number of NG bosons plus 1. From the mathematical structure of
${\cal M}$, we find that the matrix can be expressed by a Borel subalgebra (
given by the direct sum of Cartan subalgebra ${\bf h}$ and the positive-weight
part of ${\bf g}$ ) of ${\rm Lie}GL(N+1)$ or ${\rm Lie}O(N+1)$ [43]. Probably,
we can employ a flag variety to study a representation of such an expression
of Borel subalgebra, Borel subgroup $B$, the Borel-Weil theorem, and a Bruhat
decomposition of $G=\cup_{\omega\in W}B\omega B\ni g$ as a disjoint union (
$G$; a connected reductive algebraic group, $W$; a Weyl group ): The mode-mode
coupling part of the matrix ${\cal M}$ is decomposed into a strictly upper
triangular matrix and a strictly lower triangular matrix, and they can be
expressed by a Borel subalgbera. ( Note: Any strictly triangular matrix is
nilpotent, and a set of strictly upper/lower triangular matrices forms a
nilpotent Lie algebra. Borel subgroup = invertible triangular matrices = all
diagonal entries must be nonzero. Borel subalgebra = not necessarily
invertible. ) Of particular interest is a homogeneous space $G/P=BWB/P$ ( $P$;
a parabolic subgroup ) which gives a parabolic geometry. Since ${\cal
L}\in{\rm Lie}O(N+1)$ and the local coordinate system ( NG bosons ) are all
real, one can consider an adjoint group action $g^{-1}{\cal L}g$ ( $g\in
O(N+1)$ ). The equation ${\rm det}{\cal M}(p_{\nu}\neq 0)=0$ defines a
hypersurface in such a Borel subgroup representation space. The Borel-Weil
theorem states that the global section $\Gamma(G/B,L_{\lambda})$ (
$L_{\lambda}$ is a $G$-equivariant holomorphic line bundle over $G/B$,
$\lambda$ denotes a dominant integral highest weight ) gives an irreducible
representation of $G$. Hence, the mechanism of generating massive NG bosons
reflects a breaking of $O(N+1)$ ( more precisely, $O(N+1,{\bf R})$ ) symmetry
as a result of a breaking scheme and a Lorentz-violating parameter, and the
breaking structure of $O(N+1)$ reflects the dispersion relations and numbers
of massive/massless NG bosons. We will see later that this $O(N+1)$ structure
gives a symmetry ( in fact, degeneracy ) of the mass spectrum of NG bosons: In
fact, as we will see at (116) of the decomposition into several Heisenberg
pairs, the number and structure of Heisenberg pairs embedded in ${\cal L}$ is
determined by the $O(N+1)$ structure. When we observe our Lagrangian from the
context of chiral perturbation theory, the expansion of a chiral perturbation
may be given in a form of a series of symplectic matrices ( a series of the
form proportional to $\sum_{n}c_{n}(1+c_{A}T^{A})^{n}$, $T^{A}\in{\rm Lie}(G)$
).

Our result and discussion given here depends on our observation of the kaon
condensation model with the general theory of effective action/potential, and
thus we cannot say definitely whether there is another mechanism ( different
from the mode-mode coupling mechanism ) which gives a similar
phenomenon/result of anomalous NG theorem at a Lagrangian level. Our result
given here is derived from a relativistic model, though it can be applied to
the cases of (anti)ferromagnets, since their low energy excitations are
described by the class of $O(3)$ ( and mathematically generally, $O(N)$ )
nonlinear sigma models. In that case, according to our examination of the
Heisenberg algebra coming from $SU(2)$, mode-mode coupling terms similar to
the case of relativistic model with a finite chemical potential may be
introduced in the sigma model of a nonrelativistic case. Therefore, if such
type of nonlinear sigma models can be derived from Ginzburg-Landau-Goldstone-
Higgs-Kibble-type theories, then similar phenomenon may be observed. In fact,
the Lagrangian of ferromagnet given in the paper of Watanabe and Murayama [82]
takes almost a special example of our generic Lagrangian (104). Namely,

Theorem: The mechanism of anomalous NG theorem in a nonrelativistic case is
the same with that of a Lorentz-violating relativistic case. The counting law
of the number of NG bosons of a nonrelativistic case is the same with that of
a Lorentz-violating relativistic case.

It is interesting for us to study a phase of magnon condensation by our
theoretical framework presented here, since a condition of Bose-Einstein
condensation is determined by a mass parameter and a chemical potential ( a
magnon energy in a real substance is typically $\mu$eV ).

Our generic Lagrangian given above can be discussed more systematically and
mathematically/geometrically [42]. With respect to the procedure to obtain the
quadratic part in terms of NG bosons from the Lagrangian of kaon condensation,
a kinetic part is expressed as follows:

$\displaystyle\Phi$ $\displaystyle=$ $\displaystyle g\Phi_{0},\qquad
g=e^{iQ^{A}\chi^{A}}\in G,$ (113)
$\displaystyle\tilde{\partial}^{\dagger}_{\nu}\Phi^{\dagger}\tilde{\partial}^{\nu}\Phi$
$\displaystyle=$
$\displaystyle-\Phi^{\dagger}_{0}(g^{-1}\tilde{\partial}_{\nu}g)(g^{-1}\tilde{\partial}^{\nu}g)\Phi_{0},$
(114) $\displaystyle=$
$\displaystyle-\Phi^{\dagger}_{0}\Bigl{(}g^{-1}(\partial_{\nu}\partial^{\nu}-2i\mu\partial_{0}-\mu^{2})g\Bigr{)}\Phi_{0}.$

Here, we have chosen the local coordinate system $\\{\chi^{A}\\}$ as the first
kind. The term of $2i\mu\partial_{0}$ is coming from the Leibniz rule of
derivation. In general, the kinetic term is given by
$g_{\alpha\bar{\beta}}\tilde{\partial}^{\dagger}_{\nu}(\Phi^{\dagger})^{\beta}\tilde{\partial}^{\nu}(\Phi)^{\alpha}$,
and $\tilde{g}=g_{\alpha\bar{\beta}}dz^{\alpha}\otimes d\bar{z}^{\beta}$ (
$dz^{\alpha}\in{\bf C}$, $d\bar{z}^{\beta}\in\bar{\bf C}$ ) gives a Hermitian
metric. This Lagrangian does not have the fluctuation of amplitude mode of
$\Phi$, which can couple with an NG boson as we have observed in the kaon
condensation model. In fact, the amplitude mode cannot be expressed by a
Maurer-Cartan form. It should be noticed that,

$\displaystyle g^{-1}\tilde{\partial}_{0}g=g^{-1}\partial_{0}g-i\mu.$ (115)

Note that this equation shows a ”deformation” of ( or, a deviation from ) the
Maurer-Cartan 1-form

$\displaystyle\omega(T^{A},\theta^{A})=g^{-1}\partial_{\nu}g=i\sum
T^{A}\otimes\theta^{A},\quad\theta^{A}=\partial_{\nu}\chi^{A},$ (116)

as a geometric object ( $\\{T^{A}\\}$ give the Maurer-Cartan frame,
$\\{\theta^{A}\\}$ are the Maurer-Cartan coframe ), or a deviation from the
Killing form given in terms of $g^{-1}\partial_{\nu}g$, namely ${\rm
tr}(g^{-1}\partial_{\nu}g,g^{-1}\partial^{\nu}g)$ in the Lagrangian. It must
be noticed that $g^{-1}\partial_{0}g\in{\rm Lie}(G)\simeq T_{e}G$ ( $e$: the
origin ), while the part $-i\mu$ is independent from a geometric structure of
${\rm Lie}(G)$. A Maurer-Cartan form is an adjoint orbit, defines essentially
a homogeneous space. Moreover, a Maurer-Cartan form defines a local section: A
vacuum state of the quantum field theory is given by the specific local
section of $V_{eff}$ in the sense of fiber bundle, and a structural group (
namely, a Lie group ) gives a transformation between two local sections. The
effective potential $V_{eff}$ is an example of so-called representation
function: $V_{eff}$ is a continuous function defined over a topological space
$X$, and has a continuous group action of a group element of $G$. Thus, the
group orbit is defined by the pair $(V_{eff},g\in G)$, which is a subset of
the space of continuous functions over $X$. In other words, $V_{eff}$ belongs
to a set of continuous sections of a $G$-bundle $E$. Namely,
$V_{eff}\in\Gamma(E)_{G}\subset\Gamma(E)$, where $\Gamma(E)$ is the space of
all continuous sections, $\Gamma(E)_{G}$ is its submodule. $\Gamma(E)_{G}$ is
dense in $\Gamma(E)$ when $G$ is compact. This fact is important to certify a
variational calculus of $V_{eff}$ to obtain a stationary point. Note that the
Killing form is now subjected to the Euler-Lagrange variation principle, and
of course defines a phase factor of path integral. In the usual case, a
connection $\xi$ is introduced by the form $g^{-1}(d+\xi)g$, and the part
$g^{-1}dg$ is the Maurer-Cartan form, namely the chemical potential is
included in our theory under the manner of a connection. Note that in our
case, $\chi^{A}$ have the additional dependence on spacetime coordinates
coming from the local ”wave” nature of quantum field theory which is not
contained in the traditional Lie theory. From our results, we know the mode-
mode couplings of NG bosons are given as VEVs of the space of Lie algebra of
the 1-form, such as $g^{-1}\mu(\partial_{0}\chi^{A})Q^{A}g$, namely a
displacement at the origin caused by a Lie group in the derivative of NG
boson, and the explicit symmetry breaking parameter $\mu$ acts on them as a
linear scaling factor. Moreover, any deformation theory of mathematics of
manifolds defined over a field of characteristic zero is described by Maurer-
Cartan elements of a differential graded Lie algebra ( for example, the theory
of deformation quantization of Kontsevich [45] ). In our case, since our
anomalous NG theorem quite often gives a ”quasi” Heisenberg algebra over a
Poisson manifold, from locally to globally, thus, we need a special study on a
deformation theory and a deformation quantization which can achieve a quasi-
Heisenberg algebra/group. In addition, our $\omega$ satisfies the Maurer-
Cartan structural equation, sometimes called as the deformation equation,

$\displaystyle
d\omega(T^{A},\theta^{A})+\frac{1}{2}[\omega(T^{A},\theta^{A}),\omega(T^{A},\theta^{A})]=0,\quad{\rm
or},$ (117) $\displaystyle
d\theta^{A}+i\sum_{B,C}f^{BCA}\theta^{B}\wedge\theta^{C}=0.$ (118)

The Maurer-Cartan equation is the vanishing condition of the curvature 2-form,
namely the vanishing condition of the curvature of Cartan connection
$\Omega=d\omega+\frac{1}{2}[\omega,\omega]=0$. ( Understood as Maurer-Cartan
form $\subset$ Cartan connection. ) The Maurer-Cartan equation always holds
for any $\omega$. This definition of curvature is independent from the notion
of ”curvature” we use in the second-order derivative of the effective
potential $V_{eff}$: In fact, $V_{eff}$ is a function of local coordinates,
while a transition function between two local coordinate systems over
$V_{eff}$ is determined by the geometric structure of $V_{eff}$ itself, thus
the transition function cannot be defined group theoretically a priori (
mostly, $GL(n,{\bf R})$ ). The symmetry in the vicinity of a point defined
over $V_{eff}$ can be found from the curvature matrix as the second order
derivative of $V_{eff}$ since the mass matrix of NG sector reflects the
symmetry at a point. If the mass spectrum of NG sector has a degeneracy, then
the symmetry at the point becomes higher. Now the Cartan connection is an
affine connection of frame bundle ( i.e., a tangent bundle ) of the base
manifold, and also be interpreted as a special example of connection of
principal bundle. Due to the Cartan-Ambrose-Hicks theorem, a manifold is
locally Riemannian symmetric if and only if its curvature is constant, and for
any simply connected, complete locally symmetric space is Riemannian
symmetric. The expressions of a Lagrangian given in terms of $\omega=g^{-1}dg$
are familiar in theory of nonlinear realization [49]: The reason is needless
to say. For a case of a Clifford-Klein form $\Gamma\backslash G/H$ of a
symmetric space, the following parametrization will be introduced:

$\displaystyle\Phi$ $\displaystyle=$ $\displaystyle\gamma
gh\Phi_{0},\quad\gamma\in\Gamma,\,g\in G,\,h\in H.$ (119)

Hence, a lattice element $\gamma$ is contained in the Maurer-Cartan form
$(\gamma gh)^{-1}d(\gamma gh)$. Here, as we have mentioned above, a lattice
$\gamma$ gives a symmetry of a set of stationary points of $V_{eff}$ over
$G/H$, and thus $\gamma$ implies an equivalence of physics of $(gh)^{-1}d(gh)$
and $(\gamma gh)^{-1}d(\gamma gh)$ in an NG boson Lagrangian ( a kind of
replication of a theory takes place ). Due to involutions and $\gamma$, the
global structure of the NG manifold in a symmetric space is well-
characterized. The part of mode-mode couplings in our Lagrangian is coming
from the cross terms of a product of similarity transformations in the above
expression (108). In fact, these expressions have the advantage to consider
some geometric nature of the anomalous NG theorem, since the manner of how a
Lorentz-symmetry-violating parameter couples with an internal symmetry of Lie
group will be explained by the words of geometry. Note that $\partial_{\nu}$
are generators of subalgebra of Poincaré algebra. It should be mentioned that,
when $\Phi$ is an order-parameter of Dirac mass type which spontaneously
breaks a chiral $\gamma_{5}$ symmetry ( coming from an underlying fermion
system ), a transform $\Phi\to e^{2i\gamma_{5}\theta_{5}}\Phi$ can be
considered in the Lagrangian, though it does not give a mode-mode coupling
between NG bosonic fields inside the Lagrangian. Hence, at least in our
definition of bosonic Lagrangian, the structure of mode-mode couplings of NG
bosons we have revealed in this paper is closed inside the internal symmetry
of a Lie group. ( Or, can say an internal symmetry can couple with a Poincaré
generator $\partial_{\nu}$ but cannot couple with the $\gamma_{5}$-rotation. )
On the other hand, there are couplings between the chiral
$\gamma_{5}$-symmetry and NG bosons in an explicit+dynamical symmetry breaking
in an NJL-type $SU(N)$ model with $N>2$ ( see [70] ). In the latter case, the
$\gamma_{5}$-chiral symmetry and the flavor symmetry are coupled with each
other through the chiral projection operator
$P_{\pm}=\frac{1\pm\gamma_{5}}{2}$. In Ref. [70], the author discussed that a
chiral mass can be handled as a Riemann surface. Since a
$\gamma_{5}$-transformation is vertical with the internal symmetry of $SU(2)$,
$\Phi$ gives a direct sum of Riemann surfaces. ( In the case of $SU(N)$ with
the case when $\Phi$ takes a fundamental representation ${\bf N}$ or
$\overline{\bf N}$, a direct sum of $N$ Riemann surfaces will be given. ) In
such a case, since some NG bosons become massive and the effective potential
$V_{eff}$ is periodic with respect to the directions of massive modes in our
anomalous NG theorem, some Galois-group symmetries of both $\gamma_{5}$ and
massive modes arise simultaneously in a theory. In other words, the theory
gives a Galois representation of those degrees of freedom ( $\gamma_{5}$,
massive modes ) simultaneously.

Toward the understanding of counting law of massive NG bosons, we will examine
the kinetic term of bosonic Lagrangian given above, especially by using the
words of Cartan geometry. The relevant part of mode-mode couplings of bosonic
fields is obtained from the Lie-algebra expansion of the Maurer-Cartan 1-form
$g^{-1}\partial_{0}g$, namely,

$\displaystyle\Phi^{\dagger}_{0}\Bigl{\\{}e^{-i\sum_{A}Q^{A}\chi^{A}}2i\mu\sum_{B}Q^{B}(\partial_{0}\chi^{B})e^{i\sum_{A}Q^{A}\chi^{A}}\Bigr{\\}}\Phi_{0}$
(120)

Here we have used the fact that $\Phi_{0}$ is a constant ( VEV ), has no
dependence on spacetime coordinates. This term can arise only by the presence
of the Lorentz-symmetry-violating parameter $\mu$ in the Lagrangian. After
expanding the exponential mappings and picking up the quadratic terms of
bosonic fields, we get

$\displaystyle(114)$ $\displaystyle=$
$\displaystyle-2\mu\sum_{A}\sum_{B}\chi^{A}(\partial_{0}\chi^{B})\Phi^{\dagger}_{0}[Q^{B},Q^{A}]\Phi_{0}$
(121) $\displaystyle=$ $\displaystyle
2\mu{\sum\sum}_{A>B}\Bigl{\\{}\chi^{A}(\partial_{0}\chi^{B})-\chi^{B}(\partial_{0}\chi^{A})\Bigr{\\}}\Phi^{\dagger}_{0}[Q^{A},Q^{B}]\Phi_{0}.$

This is a sum of off-diagonal part ( upper triangular part without the
diagonal elements ) of matrix elements with indices $A,B$. The number of non-
vanishing terms of this sum, namely the part
$\sum_{A>B}\Phi^{\dagger}_{0}[Q^{A},Q^{B}]\Phi_{0}=i\sum_{A>B}f^{ABC}\Phi^{\dagger}_{0}Q^{C}\Phi_{0}$,
counts the number of pairs of two modes they are coupled with each other.
Maximally the number of terms is $N(N-1)/2$ when $N={\rm dim}{\rm Lie}(G)$,
and thus the number coincides with the number of generators of $SO(N)$: Hence
the linear space of mode-mode coupling matrix can be expanded by the basis set
of Lie$(SO(N))$. Moreover, each term of this sum has a correspondence with the
Lie algebra valued linear equations given in our discussion by using the
second-order derivatives of the effective potential $V_{eff}$, namely Eqs.
(68), (72) and (75). The symplectic structure discussed in Ref. [81] is
obvious in our expression (115) due to the anti-symmetric nature of structural
constants $f^{ABC}$. Thus, our Lagrangian defines a symplectic manifold with
an appropriately defined symplectic structure. A linear transformation in the
matrix space of our Lagrangian gives an isomorphism of the symplectic
structure, possiblly continuously. Moreover, if a symplectic transformation of
our Lagrangian is given over a symplectic manifold, then it is expressed as a
symplectic Lie group. For example, in the case of $SU(2)\to U(1)$ of a
ferromagnet with broken generators $(Q^{1},Q^{2})$ and a symmetric generator
$Q^{3}$, only $if^{123}\Phi^{\dagger}_{0}Q^{3}\Phi_{0}$ remains to give a
finite VEV. If we choose the representation that gives $Q^{3}$ in the form of
diagonal matrix such as the Pauli matrix $\sigma^{3}$, then this term remains
when the VEV of $\Phi$ takes the form $\Phi_{0}=(v_{1},v_{2})$ ( $v_{1}\neq
v_{2}$ ). In this case, the number of pair of bosonic fields they are coupled
is 1. When the VEVs of $[Q^{A},Q^{B}]$ are pairwise decoupled, then the set of
VEVs gives a set of Heisenberg algebras, and then the mode-mode couplings in
the space of NG bosons $(\chi_{1},\cdots,\chi_{N})$ are also decoupled into
subspaces pairwisely, and the problem of the matrix is reduced into the direct
sum of $2\times 2$ matrices ( i.e., block-diagonalized ) such that

$\displaystyle\left(\begin{array}[]{cc}k^{2}&2ic_{1}\mu k_{0}\\\ -2ic_{1}\mu
k_{0}&k^{2}\end{array}\right)\oplus\cdots\oplus\left(\begin{array}[]{cc}k^{2}&2ic_{l}\mu
k_{0}\\\ -2ic_{l}\mu k_{0}&k^{2}\end{array}\right).$ (126)

In such a case, our discussion on dispersion relations of NG bosons is reduced
very much. The decomposition to $2\times 2$ matrices can also be interpreted
as the result that a Lie algebra is constructed by the fundamental unit
$sl_{2}$-triple, apparent from a Cartan decomposition. This is the origin of
the counting law of Watanabe, Brauner, and Hidaka: It is obvious from our
analysis, rank$\langle[X^{A},X^{B}]\rangle$ counts the number of pairs of
mode-mode coupling, and thus it can estimate the dimension of a matrix of non-
vanishing mode-mode coupling elements, though the pairwise decoupling must
take place to conclude definitely that the number of massive NG bosons is the
half of the rank. The cases of

$\displaystyle\langle h_{i}\rangle\neq 0,\quad\langle e_{j}\rangle\neq
0,\quad\langle f_{k}\rangle=0,$ (127)

and

$\displaystyle\langle h_{i}\rangle\neq 0,\quad\langle e_{j}\rangle\neq
0,\quad\langle f_{k}\rangle\neq 0,$ (128)

give more complicated situation for the counting law.

The adjoint orbit $O(X)={\rm Ad}(G)X$ ( $X\in{\rm Lie}(G)$ ) defines a
submanifold of ${\rm Lie}(G)$. A typical example is the Maurer-Cartan form
$g^{-1}dg$, and the curvature 2-form will be defines as a function of the
adjoint orbit. Needless to say, an NG manifold consists of adjoint orbits:

$\displaystyle O(\Phi)$ $\displaystyle=$ $\displaystyle g^{-1}\Phi g\simeq
e^{i\chi^{A}X^{A}}\Phi,\quad\Phi\in{\rm Lie}(G).$ (129)

After choosing the specific form of $\Phi\in{\rm Lie}(G)$ ( for example, when
$\Phi$ takes its value in the Cartan subalgebra of ${\rm Lie}(G)$ ),
$O(\Phi)=g^{-1}\Phi g$ gives a homogeneous space $G/G(\Phi)$, where $G(\Phi)$
is the stabilizer $\\{g\in G|{\rm Ad}(g)\Phi=\Phi\\}$. If $G$ is compact, an
adjoint orbit is called as an elliptic orbit.

Especially the case of $SU(2)\to U(1)$ gives a more explicit geometric
interpretation. Let us consider a situation where a one-dimensional curve is
defined over a two-dimensional oriented surface, and the surface is embedded
into a Euclidean space ${\bf R}^{3}$. Let ${\bf T}$ be the unit tangent vector
of the curve, let ${\bf N}$ be the unit normal vector of the surface, and
${\bf S=N\times T}$ is the tangent normal. The ${\bf T}$, ${\bf N}$ and ${\bf
S}$ gives the Darboux frame, an orthogonal frame. Let $C_{normal}$ be the
normal curvature, $C_{geodesic}$ be the geodesic curvature, and let
$T_{geodesic}$ be the geodesic torsion. It is a known fact that those
quantities are given by the following linear transformation ( the Frenet-
Serret equation ):

$\frac{d}{ds}\left(\begin{array}[]{c}{\bf T}\\\ {\bf S}\\\ {\bf
N}\end{array}\right)=\left(\begin{array}[]{ccc}0&C_{geodesic}&C_{normal}\\\
-C_{geodesic}&0&T_{geodesic}\\\
-C_{normal}&-T_{geodesic}&0\end{array}\right)\left(\begin{array}[]{c}{\bf
T}\\\ {\bf S}\\\ {\bf N}\end{array}\right).$ (130)

Apparently, the matrix of this linear transformation is a group element of
$SO(3)$. Hence the off-diagonal part of our Lagrangian of the anomalous NG
theorem in the $SU(2)$ case has this geometric implication, a curve on a
surface in ${\bf R}^{3}$. An important difference is that ${\bf S}$ is defined
by ${\bf T}$ and ${\bf N}$, while $(\chi_{1},\chi_{2},\chi_{3})$ are linearly
independent with each other: Namely, the projective case
$\chi^{2}_{1}+\chi^{2}_{2}+\chi^{2}_{3}=1$ corresponds to the Frenet-Serret
equation. A matrix element of the off-diagonal part corresponds to a curvature
or a torsion, and the local coordinates of $SU(2)$ ( i.e., the NG bosons )
gives an orthonormal frame. In other words, $[Q^{A},Q^{B}]$ define the local
geometry of the NG manifold. Those mathematical structure may be hidden in the