For any *B* ∈ R, the quantum Ising chain has a Z2-symmetry given by a 180 degree rotation around the *x*-axis, locally implemented by the unitary operator *u*(*x*) = σ1(*x*), which at each *x* ∈ Λ*N* yields (σ1,σ2,σ3) → (σ1,−σ2,−σ3), since σ*i*σ*j*σ∗ *i* = −σ*j* for *i* = *j*. Thus *u*(*x*) sends each σ3(*x*) to −σ3(*x*) but maps each σ1(*x*) to itself. As in (10.57), this symmetry is implemented by the unitary operator

$$\mu^{(N)} = \copysim\_{\mathfrak{x} \in \mathsf{A}\_N} \mathsf{σ}\_1(\mathfrak{x}) \tag{10.68}$$

on *HN*, which satisfies [*hN*,*u*(*N*) ] = 0, or, equivalently,

$$
u^{(N)}h\_N(\
u^{(N)})^\* = h\_N.\tag{10.69}$$

The ensuing Z2-symmetry is given by the automorphism γ(*N*) of *AN* defined by

$$\gamma^{(N)}(a) = \mu^{(N)} a (\mu^{(N)})^\* \ (a \in A\_N), \tag{10.70}$$

which induces a global automorphism γ ∈ Aut(*A*) as in (10.59), i.e.,

$$\mathcal{Y}(a) = \lim\_{N \to \infty} u^{(N)} a(u^{(N)})^\* \ (a \in A), \tag{10.71}$$

which limit once again exists despite the fact that the sequence *u*(*N*) has no limit in *A*. Thus Z2-invariance of the model follows from the local property

$$
\alpha\_{\mathfrak{r}}^{(N)} \circ \mathfrak{Y}^{(N)} = \mathfrak{Y}^{(N)} \circ \mathfrak{a}\_{\mathfrak{r}}^{(N)},\tag{10.72}
$$

which in the limit *N* → ∞ gives

$$
\alpha \mathfrak{a} \circ \lambda = \mathfrak{y} \circ \mathfrak{a} \quad (\mathfrak{r} \in \mathbb{R}).\tag{10.73}
$$

Since γ2 = id*A*, we have an action of the group Z2 = {−1,1} on *A*, where the nontrivial element (i.e., *g* = −1) is sent to γ. By (10.72) this group acts on the set *S*∞(*AN*) of ground states of *AN* relative to the dynamics α(*N*) , and by (10.73) the same is true for the set *S*∞(*A*) of ground states of the corresponding infinite system for α (and analogously for β-KMS states). These sets may be described as follows.

$$a\_0^{(0)} = \,\_2^!(a\_0^+ + a\_0^-) \tag{10.74}$$

*form a continuous field of states on the continuous bundle A*(*q*) *; in particular,*

$$\lim\_{N \to \infty} \mathfrak{o}\_{1/N}^{(0)} = \mathfrak{o}\_0^{(0)}.\tag{10.75}$$

The two ground states in no. 1 and no. 3 are tensor products of | ↑ and | ↓, respectively (where σ3| ↑ = | ↑ and σ3| ↓ = −| ↓), so that σ3(0) is an order parameter in the sense of Definition 10.6. In no. 4, on the other hand, each spin aligns with the magnetic field in the *x*-direction, so that the ground state is an infinite tensor product of states | →, where σ1| → = | →, and this time σ1(0) is an order parameter.

Case no. 2 becomes more transparent if we realize the Hilbert space *HN* as -2(*SN*), where *SN* is the set of all spin configurations *s* on *N* sites, that is,

$$s: \{ -\frac{1}{2}N, -\frac{1}{2}N+1, \dots, \frac{1}{2}N-1 \} \to \{ -1, 1 \}.$$

In terms of the eigenvectors |1 ≡ | ↑ and | −1 ≡ | ↓ of σ3, and the orthonormal basis (δ*s*)*s*∈*SN* of -2(*SN*) (where δ*s*(*t*) = δ*st*), a suitable unitary equivalence

$$\nu\_N: \ell^2(\mathbb{S}\_N) \to H\_N \tag{10.76}$$

is given by linear extension of

$$\mathbf{v}\_{N}\mathbf{\hat{o}}\_{s} = |\mathbf{s}(-\frac{1}{2}N)\cdots\mathbf{s}(\frac{1}{2}N-1)\rangle,\ \mathbf{s},t \in \mathbb{S}\_{N}.\tag{10.77}$$

For example, the state |1···1 corresponds to δ*s*↑ , where *s*↑(*x*) = 1 for all *x*, and analogously *s*↓(*x*) = −1 for the state | − 1···− 1. Using -2(*SN*), we may talk of localization of states in spin configuration space (similar to localization of wavefunctions in *L*2(R*n*)), in the sense that some ψ ∈ -2(*SN*) may be peaked on just a few spins configurations. Provided 0 < *B* < 1 this is indeed the case for the unique ground state in case no. 2, which is similar to the ground state of the double-well potential discussed in §§10.1–10.2, replacing R by *SN* (and *h*¯ > 0 by 1/*N*).

Theorem 10.11 and related results used below, such as eq. (10.82), follow from the exact solution of the model for both *N* < ∞ and *N* = ∞, to be discussed in §§10.6–10.7. This solution is rather involved, but a rough picture of the various ground states may already be obtained from a classical approximation in the spirit of §8.1. This approximation assumes that the spin-1/2 operators 1 2σ*i* are replaced by their counterparts for spin *n* · 1 2 , upon which one takes the limit *n* → ∞. In this limit, the spin operators are turned into the corresponding coordinate functions on the coadjoint orbit O1/2 ⊂ R3 for *SU*(2), which is the two-sphere *S*2 1/2 with radius *r* = 1/2. In principle, this should be done for each of the *N* spins separately, yielding a classical Hamiltonian *hc* that is a function on the *N*-fold cartesian product of *S*2 1/2 with itself. However, if we *a priori* assume translation invariance of the classical ground state, only one such copy remains. Using spherical coordinates

$$\theta \left( x = \frac{1}{2} \sin \theta \cos \phi, y = \frac{1}{2} \sin \theta \sin \phi, z = \frac{1}{2} \cos \theta \right), \tag{10.78}$$

the ensuing trial Hamiltonian becomes just a function on O1/2, given by

$$h(\theta, \phi) \approx -(\frac{1}{2}\cos^2\theta + B\sin\theta\cos\phi). \tag{10.79}$$

Minimizing gives cosφ = 1 and hence *y* = 0 for any *B*, upon which

$$h(\theta) \approx -\left(\frac{1}{2}\cos^2\theta + B\sin\theta\right) \tag{10.80}$$

yields the phase portrait of Theorem 10.11 for *N* = ∞, as follows. For 0 ≤ *B* < 1, the global minimum is reached at the two different solutions θ± of cosθ± = *B*, with ensuing spin vectors

$$\mathbf{x}\_{\pm}(\mathcal{B}) = (\frac{1}{2}\mathcal{B}, 0, \pm\frac{1}{2}\sqrt{1 - \mathcal{B}^2}),\tag{10.81}$$

starting at x±(0)=(0,0,±1 2 ) and merging at *B* = 1 to x+(1) = x−(1)=( 1 2 ,0,0). This remains the unique ground state for *B* ≥ 1, where all spins align with the field.

In the regime 0 < *B* < 1 with large but finite *N*, one finds a far-reaching analogy between the double-well potential and and the quantum Ising chain, namely:

$$
\Delta\_N \approx (1 - B^2) B^N \ (N \to \ast \ast), \tag{10.82}
$$

for the quantum Ising chain. Thus both models show exponential decay, i.e. of (10.82) in *N* as *N* → ∞, and of (10.13) in 1/*h*¯ as *h*¯ → 0.

It should be mentioned that *exponential* decay of the energy gap seems a lowdimensional luxury, which is not really needed for SSB. All that counts is that lim*N*→∞ Δ*N* = 0, which guarantees that the first excited state is asymptotically degenerate with the ground state, so that appropriate linear combinations like ω± 0 can be formed that converge to the degenerate symmetry-breaking *pure* (and hence physical) ground states (or *extreme* and hence physical KMS states) of the limit system, which are localized and stable (as is clear from the double well). The fact that in the two models at hand only *one* excited state participates in this mechanism is due to the simple Z2 symmetry that is being broken; SSB of continuous symmetries requires a large number of low-lying states that are asymptotically degenerate with the ground state and hence also with each other—one speaks of a *thin* energy spectrum).

The existence of low-lying excited states may be proved abstractly (i.e., in a model-independent way), as follows. For *N* < ∞, let ψ(0) *N* be the ground state (assumed unique) of some model defined on Λ*N* ⊂ Z*d*, and let φ be an order parameter (cf. Theorem 10.7) with accompanying vector field Φ*N* = ∑*x*∈Λ*N* φ(*x*); in the quantum Ising chain, we take φ = σ1. Then the key assumptions are expressed by

$$
\langle \Psi\_N^{(0)}, \Phi\_N \Psi\_N^{(0)} \rangle = 0; \tag{10.83}
$$

$$<\langle \Phi\_N \Psi\_N^{(0)}, \Phi\_N \Psi\_N^{(0)} \rangle \ge \mathcal{C}\_1 \cdot N^2 \ (N \to \ast, \mathcal{C}\_1 > 0);\tag{10.84}$$

$$\left| \left| \left[ [\Phi\_N, h\_N], \Phi\_N \right] \right| \right| \le C\_2 \cdot N \text{ ( $N \to \approx$ ,  $C\_2 > 0$ )}.\tag{10.85}$$

The first states that the ground state is symmetric, the second enforces long-range order, as in (10.41), and the third follows from having short-range forces. A simple computation then shows that the unit vector ψ˜ (1) *N* = Φ*N*ψ(0) *N* /Φ*N*ψ(0) *N* satisfies

$$
\langle \Psi\_N^{(1)}, h\_N \Psi\_N^{(1)} \rangle - \langle \Psi\_N^{(0)}, h\_N \Psi\_N^{(0)} \rangle \le C\_2 / \langle C\_1 N \rangle \ (N \to \infty). \tag{10.86}
$$

Since ψ˜ (1) *N* is orthogonal to ψ(0) *N* by (10.83), the variational principle for eigenvalues (note that *hN* has discrete spectrum, as dim(*H*Λ*N* ) < ∞) then gives Δ*N* ≤ *C*2/(*C*1*N*), so that Δ*N* vanishes as *N* → ∞, though perhaps not as quickly as (10.82) indicates.

#### 10.6 Exact solution of the quantum Ising chain: *N* < ∞

The solution of the quantum Ising chain is based on a transformation to fermionic variables. Let *H* be a Hilbert space and let *F*−(*H*) be its *fermionic Fock space*, i.e.,

$$F\_{-}(H) = \oplus\_{k=0}^{\infty} H\_{-}^{k},\tag{10.87}$$

where *H*0 = C, and for *k* > 0 the Hilbert space *Hk* − = *e* (*k*) − *Hk* is the totally antisymmetrized *k*-fold tensor product of *H* with itself, see also §7.7. Here the projection *e* (*k*) − : *Hk* → *Hk* is defined by linear extension of

$$e\_{-}^{(k)}f\_{1}\otimes\cdots\otimes f\_{k} = \frac{1}{k!} \sum\_{p\in\mathfrak{S}\_{k}} \text{sgn}(p) f\_{p(1)}\otimes\cdots\otimes f\_{p(k)},\tag{10.88}$$

where S*k* is the permutation group on *k* objects, and sgn(*p*) is +1/ − 1 if *p* is an even/odd permutation. With the (total) Fock space *F*(*H*) = ⊕∞ *k*=0*Hk* we have *F*−(*H*) = *e*−*F*(*H*), where *e* = ∑*k e* (*k*) − (strongly) is a projection. For *f* ∈ *H* we define the (unbounded) *annihilation operator a*(*f*) on *F*(*H*) by (finite) linear extension of

$$a(f)f\_1 \otimes \cdots \otimes f\_k = \sqrt{k} \langle f, f\_1 \rangle\_H \otimes \cdots \otimes f\_k,\tag{10.89}$$

for *k* > 0, with *a*(*f*)*z* = 0 on *H*0 = C. This gives the adjoint *a*(*f*)∗ ≡ *a*∗(*f*) as

$$a^\*(f)f\_1 \otimes \cdots \otimes f\_k = \sqrt{k+1}f \otimes f\_1 \otimes \cdots \otimes f\_k. \tag{10.90}$$

For each *f* ∈ *H*, we then define the following operators on *F*−(*H*):

$$c(f) = e\_-a(f)e\_-;\tag{10.91}$$

$$c^\*(f) = e\\_{a^\*}(f)e\\_{.} \tag{10.92}$$

Note that the map *f* → *c*(*f*) is *antilinear* in *f* , whereas *f* → *a*∗(*f*) is *linear* in *f* . It follows that *c*∗(*f*) = *c*(*f*)∗, that each operator *c*(*f*) and *c*(*f*) on *F*−(*H*) is bounded with *c*(*f*) = *c*∗(*f*) = *f* , and the *canonical anticommutation relations* hold:

$$[c(f), c^\*(g)]\_+ = \langle f, g \rangle\_{H} \cdot 1\_{F\_{-}(H)};\tag{10.93}$$

$$[c(f), c(g)]\_{+} = [c^\*(f), c^\*(g)]\_{+} = 0. \tag{10.94}$$

Thus we may define CAR(*H*) as the C\*-algebra within *B*(*F*−(*H*)) generated by all *c*(*f*), where *f* ∈ *H*. This is called the C\*-algebra of *canonical anticommutation relations* over *H*, which have constructed in its defining representation on *F*−(*H*). Choosing an orthonormal basis (*ei*) of *H* and writing *c*(*ei*) = *ci* etc. clearly yields

$$[c\_i, c\_f^\*]\_+ = \delta\_{if} \cdot 1\_{F\_-(H)};\tag{10.95}$$

$$[c\_l, c\_j]\_+ = [c\_l^\*, c\_j^\*]\_+ = 0.\tag{10.96}$$

If dim(*H*) = *N* < ∞, then CAR(*H*) = *B*(*F*−(*H*)). First, a dimension count yields

$$F\_{-}(\mathbb{C}^{N}) = \oplus\_{k=0}^{N} H\_{-}^{k} \cong \mathbb{C}^{2^{N}} \cong \otimes^{N} \mathbb{C}^{2}.\tag{10.97}$$

By Theorem C.90, the C\*-algebra CAR(*H*) acts irreducibly on *F*−(*H*), so that

$$\mathbf{CAR}(\mathbb{C}^{N}) \cong \mathbf{M}\_{2^{N}}(\mathbb{C}).\tag{10.98}$$

This is already nontrivial for *N* = 1. In that case, *F*−(C) = C⊕C = C2, and

$$c = \sigma\_- = \begin{pmatrix} 0 \ 0 \\ 1 \ 0 \end{pmatrix};\tag{10.99}$$

$$c^\* = \mathfrak{o}\_+ = \begin{pmatrix} 0 \ 1 \\ 0 \ 0 \end{pmatrix},\tag{10.100}$$

where σ± = 1 2 (σ1 ±*i*σ2). This realization explicitly shows that

$$\mathbf{CAR}(\mathbb{C}) = M\_2(\mathbb{C}).\tag{10.101}$$

To generalize this to *N* > 1, we introduce a lattice (or chain) *N* = {1,...,*N*}, and for each *x* ∈ *N* we define operators *cx*, *c*∗ *x* by the *Jordan–Wigner transformation*

$$\sigma\_{\mathbf{x}} = e^{\pi i \sum\_{\mathbf{y}=1}^{x-1} \sigma\_{+}(\mathbf{y}) \sigma\_{-}(\mathbf{y})} \sigma\_{-}(\mathbf{x}) = \left( \prod\_{\mathbf{y}=1}^{x-1} (-\sigma\_{3})(\mathbf{y}) \right) \cdot \sigma\_{-}(\mathbf{x});\tag{10.102}$$

$$c\_{\mathbf{x}}^{\*} = e^{-\pi i \sum\_{\mathbf{y}=1}^{\mathbf{x}-1} \sigma\_{+}(\mathbf{y}) \sigma\_{-}(\mathbf{y})} \sigma\_{+}(\mathbf{x}) = \left(\prod\_{\mathbf{y}=1}^{\mathbf{x}-1} (-\sigma\_{3})(\mathbf{y})\right) \cdot \sigma\_{+}(\mathbf{x}), \qquad (10.103)$$

where *x* > 1, and *c*1 = σ− 1 and *c*∗ 1 = σ+ 1 (here σ±(*x*) = 1 2 (σ1(*x*) ± *i*σ2(*x*)) etc.). These operators satisfy (10.95) - (10.96); the second expression on each line follows because the operators σ+(*y*)σ−(*y*) commute for different sites *y*, and

$$e^{\pi i \sigma^+ \sigma^-} = -\sigma\_3. \tag{10.104}$$

Furthermore, since

$$\mathbf{c}\_x^\* \mathbf{c}\_x = \sigma\_+(\mathbf{x}) \sigma\_-(\mathbf{x}) = \begin{pmatrix} 1 \ 0 \\ 0 \ 0 \end{pmatrix} (\mathbf{x});\tag{10.105}$$

$$c\_{\mathbf{x}}c\_{\mathbf{x}}^{\*} = \sigma\_{-}(\mathbf{x})\sigma\_{+}(\mathbf{x}) = \begin{pmatrix} 0 \ 0 \\ 0 \ 1 \end{pmatrix}(\mathbf{x}),\tag{10.106}$$

the inverse of the Jordan–Wigner transformation is given by

$$
\sigma\_{-}(\mathbf{x}) = e^{-\pi i \Sigma\_{\gamma=1}^{x-1} c\_{\gamma}^\* c\_{\gamma}} c\_{\mathbf{x}};\tag{10.107}
$$

$$\sigma\_{+}(\mathbf{x}) = c\_{\mathbf{x}}^{\*} e^{\pi l \Sigma\_{\mathbf{y}=1}^{x-1} c\_{\mathbf{y}}^{\*} c\_{\mathbf{y}}}.\tag{10.108}$$

We return to the quantum Ising model (10.60) with free boundary conditions, where we relabel the sites as {1,...,*N*}, as above, and change to the Hamiltonian

$$h\_N^{\rm QI} = -\frac{1}{2} \left( \sum\_{\mathbf{x}=1}^{N-1} \sigma\_\mathbf{l}(\mathbf{x}) \sigma\_\mathbf{l}(\mathbf{x}+1) + \lambda \sum\_{\mathbf{x}=1}^{N} \sigma\_\mathbf{\hat{3}}(\mathbf{x}) \right), \tag{10.109}$$

where, in order to avoid notational confusion with the operator *B* in (10.111) below, we henceforth replace *B* λ. In terms of the unitary operator *u* = 1/2(12 +*i*σ2) on C2 and hence *u*(*N*) = ⊗*N x*=1*u*(*x*) on ⊗*N*C2, we have *u*(*N*) *hN*(*u*(*N*) )∗ = *h N*.

Using (10.102) - (10.103), up to an additive constant λ*N* · 1*N* we omit, we find

$$h\_N^{\rm QI} = -\sum\_{\mathbf{x}=1}^N (\lambda c\_{\mathbf{x}}^\* c\_{\mathbf{x}} + \frac{1}{2} (c\_{\mathbf{x}}^\* - c\_{\mathbf{x}})(c\_{\mathbf{x}+1}^\* + c\_{\mathbf{x}+1})),\tag{10.110}$$

so we now show how to diagonalize quadratic fermionic Hamiltonians of the type

$$h\_N = -\sum\_{\mathbf{x}, \mathbf{y} = 1}^N \left( A\_{\mathbf{x}\mathbf{y}} c\_{\mathbf{x}}^\* c\_{\mathbf{y}} + \frac{1}{2} B\_{\mathbf{x}\mathbf{y}} (c\_{\mathbf{x}}^\* c\_{\mathbf{y}}^\* - c\_{\mathbf{x}} c\_{\mathbf{y}}) \right), \tag{10.111}$$

where *A* and *B* are real *N* ×*N* matrices, with *A*∗ = *A* and *B*∗ = −*B*. Indeed, taking

$$A = \frac{1}{2}(\mathbf{S} + \mathbf{S}^\*) + \mathbf{\hat{\mathcal{A}}} \cdot \mathbf{1}\_N;\tag{10.112}$$

$$B = \frac{1}{2}(S - S^\*),\tag{10.113}$$

recovers (10.110), where *S* : C*N* → C*N* is the *shift operator*, defined by

$$Sf(\mathbf{x}) = f(\mathbf{x} + 1);\tag{10.114}$$

$$S^\*f(\mathbf{x}) = f(\mathbf{x} - \mathbf{1}).\tag{10.115}$$

By convention, *f*(*N* + 1) = *f*(0) = 0 (i.e., *S f*(*N*) = *S*∗ *f*(0) = 0 for any *f* ∈ C*N*); in terms of the standard basis (υ*x*) of C*N* we have *S*υ1 = 0 and *S*υ*x* = υ*x*−1 for *x* = {2,...,*N*}, and likewise *S*∗υ*N* = 0 and *S*υ*x* = υ*x*+1 for *x* = {1,...,*N* −1}.

The smart thing to do now turns out to be diagonalizing the 2*N* ×2*N*-matrix

$$M = \begin{pmatrix} A & B \\ -B & -A \end{pmatrix},\tag{10.116}$$

which by a unitary transformation may be brought into the simpler form

$$M' = \begin{pmatrix} \sqrt{1/2} - \sqrt{1/2} \\ \sqrt{1/2} \ \sqrt{1/2} \end{pmatrix} \begin{pmatrix} A & B \\ -B - A \end{pmatrix} \begin{pmatrix} \sqrt{1/2} & \sqrt{1/2} \\ -\sqrt{1/2} & \sqrt{1/2} \end{pmatrix} = \begin{pmatrix} 0 & C \\ C^\* & 0 \end{pmatrix},\tag{10.117}$$

where *C* = *A*+*B*. For example, for the model (10.111) we simply have

$$\mathbf{C} = \mathbf{S} + \mathbf{\hat{\mathcal{A}}} \cdot \mathbf{1}\_N. \tag{10.118}$$

The equations for the eigenvalues ε*k* and eigenvectors of *M* , i.e.,

$$M'\begin{pmatrix}\boldsymbol{\varrho}\_{k} \\ \boldsymbol{\Psi}\_{k}\end{pmatrix} = \boldsymbol{\varepsilon}\_{k} \begin{pmatrix} \boldsymbol{\varrho}\_{k} \\ \boldsymbol{\Psi}\_{k} \end{pmatrix} \tag{10.119}$$

where ϕ*k*,ψ*k* ∈ C*N*, are equivalent to both the coupled system of equations

$$C\Psi\_k = \varepsilon\_k \Phi\_k;\tag{10.120}$$

$$C^\*\mathfrak{g}\_k = \mathfrak{e}\_k \mathfrak{w}\_k;\tag{10.121}$$

$$C = A + B,\tag{10.122}$$

where the eigenvalues ε*k* are real (since *M*∗ = *M*), and to the uncoupled version

$$\mathcal{C}\mathcal{C}^\*\mathfrak{g}\_{\mathbb{k}} = \mathfrak{e}\_{\mathbb{k}}^2 \mathfrak{g}\_{\mathbb{k}};\tag{10.123}$$

$$C^\*C\Psi\_k = \mathfrak{e}\_k^2 \Psi\_k;\tag{10.124}$$

$$\text{CC}^\* = A^2 - B^2 - [A, B];\tag{10.125}$$

$$\mathbf{C}^\*\mathbf{C} = \mathbf{A}^2 - \mathbf{B}^2 + [\mathbf{A}, \mathbf{B}].\tag{10.126}$$

Without loss of generality we may (and will) assume that the ϕ*k*,ψ*k* are unit vectors in C*N*, so that the corresponding unit vector in C2*N* is (ϕ*k*,ψ*k*)/ √ 2). Furthermore, since *C* (or *M*) is a matrix with real entries and the ε*k* are real, by a suitable choice of phase we may (and will) also arrange that ϕ*k*,ψ*k* have real components. Finally, it follows from (10.120) - (10.120) that (−ϕ*k*,ψ*k*) is an eigenvector of *C* with eigenvalue −ε*k*, so that the unitary transformation *U* that diagonalizes *M* , i.e.,

$$(U')^{-1}M'U' = \begin{pmatrix} -E & 0\\ 0 & E \end{pmatrix},\tag{10.127}$$

where *E* = diag(ε1,..., ε*N*), takes the form

$$U' = \frac{1}{\sqrt{2}} \begin{pmatrix} \varphi \ -\varphi \\ \Psi \ \Psi' \end{pmatrix},\tag{10.128}$$

where ϕ is the *N* × *N* matrix (ϕ1,...,ϕ*N*), seeing each vector ϕ*i* as a column, etc. Combined with (10.117), we obtain

$$U^{-1}MU = \begin{pmatrix} -E & 0\\ 0 & E \end{pmatrix};\tag{10.129}$$

$$U = \frac{1}{2} \begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix} \cdot \begin{pmatrix} \mathfrak{g} \ -\mathfrak{g} \\ \Psi \ \Psi \end{pmatrix} = \frac{1}{2} \begin{pmatrix} \Psi + \mathfrak{g} \ \Psi - \mathfrak{g} \\ \Psi - \mathfrak{g} \ \Psi + \mathfrak{g} \end{pmatrix} \equiv \begin{pmatrix} u \ \nu \\ \nu \ u \end{pmatrix},\tag{10.130}$$

where we introduced *N* ×*N* matrices

$$
\mu = \frac{1}{2}(\Psi + \Phi);\tag{10.131}
$$

$$
\nu = \frac{1}{2}(\Psi - \Phi). \tag{10.132}
$$

Using orthonormality and completeness of both the (ϕ*k*) and the (ψ*k*), one obtains

$$
\mu^\* \mu + \nu^\* \nu = 1\_H;\tag{10.133}
$$

$$
\mu^\* \nu + \nu^\* \mu = 0; \tag{10.134}
$$

$$
\mu\mu^\* + \nu\nu^\* = \mathbb{I}\_H;\tag{10.135}
$$

$$
\mu \nu^\circ + \nu \mu^\circ = 0.\tag{10.136}
$$

Of course, *u* and *v* are far from unique, as they depend on both the ordering and the phases of the vectors ϕ*k* and ψ*k*. In partial remedy of the former ambiguity we assume that 0 ≤ ε0 ≤ ε1 ≤···≤ ε*N* (which can be arranged by a suitable ordering as well as choice of sign of the eigenvectors ϕ*k*). Towards the latter, we already agreed that both the ϕ*k* and ψ*k* are real, so that also our matrices *u* and *v* have real entries.

We now explain the purpose of diagonalizing *M* in (10.116) using *u* and *v*.

Proposition 10.12.  Let c*(*f*) *and c*∗(*f*) *be the operators* (10.91) *-* (10.92)*, satisfying the CAR* (10.93) *-* (10.94)*. Define the* Bogoliubov transformation

$$
\eta(f) = c(\mathfrak{u}f) + c^\*(\mathfrak{v}f);\tag{10.137}
$$

$$
\eta^\*(f) = c^\*(\mathfrak{u}f) + c(\mathfrak{v}f),
\tag{10.138}
$$

*which extends to a linear map* α : CAR(*H*) → CAR(*H*)*, where* η(*f*) = α(*c*(*f*)) *etc. Then* α *is a homomorphism of C\*-algebras, or, equivalently, one has the CAR*

$$[\mathfrak{q}(f), \mathfrak{q}^\*(\mathfrak{g})]\_+ = \langle f, \mathfrak{g} \rangle\_H \cdot \mathbf{1}\_H;\tag{10.139}$$

$$[\eta(f), \eta(g)]\_{+} = [\eta^\*(f), \eta^\*(g)]\_{+} = 0,\tag{10.140}$$

*iff* u *and* v *satisfy* (10.133) *-* (10.134)*, with u* u, *v* v*. Moreover,* α *is invertible (and hence defines an automorphism of* CAR(*H*)*) iff in addition* (10.135) *-* (10.136) *are valid (again with with u* u, *v* v*), in which case the inverse is*

$$c(f) = \eta(\mathfrak{u}^\* f) + \mathfrak{n}^\*(\mathfrak{v}^\* f);\tag{10.141}$$

$$c^\*(f) = \eta^\*(\mathfrak{u}^\*f) + \eta(\mathfrak{v}^\*f). \tag{10.142}$$

Note that anti-linearity of v is needed to make *f* → η(*f*) anti-linear, like *f* → *c*(*f*). With respect to a base (*ei*) of *H*, the transformations (10.137) - (10.142) reads

$$
\eta\_i = \sum\_j (\overline{u}\_{ji}c\_j + \nu\_{ji}c\_j^\*);\tag{10.143}
$$

$$
\eta\_j^\* = \sum\_j (\mu\_{j\bar{\imath}} c\_j^\* + \overline{\nu}\_{j\bar{\imath}} c\_j);\tag{10.144}
$$

$$c\_i = \sum\_j (\mu\_{ij}\eta\_j + \overline{\nu}\_{ij}\eta\_j^\*);\tag{10.145}$$

$$c\_i^\* = \sum\_j (\overline{u}\_{lj}\eta\_j^\* + \upsilon\_{lj}\eta\_j). \tag{10.146}$$

*Proof.* The proof is a straightforward computation. -

In comparison with the preceding diagonalization process, where *H* = C*N*, we notice that in this process *u* and *v* were both linear, whereas in Proposition 10.12 u is linear whereas v is antilinear. This difference is easily overcome by taking u = *u* and v = *Jv*, where *J* : C*N* → C*N* is the anti-linear map *J f*(*x*) = *f*(*x*), so that *J* is a *conjugation* in being an anti-linear map that satisfies *J*∗ = *J*−1 = *J*.

Returning to our generic Hamiltonian (10.111), a straightforward computation using (10.145) - (10.146), (10.116), (10.129), and (10.133) - (10.136) yields

$$h\_N = \sum\_{k=1}^N \varepsilon\_k \eta\_k^\* \eta\_k,\tag{10.147}$$

up to a (computable) constant, where we recall that ε*k* ≥ 0 (*k* = 1,...,*N*). Note that *hN* is still defined on the fermionic Fock space *F*−(C*N*), as *hN* is a (complicated) *quadratic* expression in the operators *ci* and *c*∗ *i* on *F*−(C*N*). The point is that (as a consequence of Proposition 10.12) the η*k* and η∗ *k* also satisfy the CAR, i.e.,

$$[\eta\_l, \eta\_j^\*]\_+ = \delta\_{lj} \cdot 1\_{F\_-(H)};\tag{10.148}$$

$$[\eta\_l, \eta\_f]\_+ = [\eta\_l^\*, \eta\_f^\*]\_+ = 0. \tag{10.149}$$

Theorem 10.13. *Let A* = CAR(C*N*) *be the CAR-algebra over H* = C*N with dynamics* α*t*(*a*) = *eithN ae*−*ithN given by* (10.111) *and hence by*(10.147)*. Then* α *has a unique (and hence pure and symmetric) ground state* ω0*, specified by the property*

$$
\pi\_{a\_0}(\eta(f))\varOmega\_{a\_0} = 0 \ (f \in \mathbb{C}^N). \tag{10.150}
$$

*Proof.* Recall that α defines a derivation δ : CAR(C*N*) → CAR(C*N*) defined by (9.54), which in the case at hand is simply by δ(*a*) = *i*[*hN*,*a*] (since *A* is finitedimensional, δ is bounded and hence defined everywhere). Using the identity

$$[ab,c] = a[b,c]\_+ - [c,a]\_+b,\tag{10.151}$$

as well as the relations (10.148) - (10.149), we obtain δ(η*k*) = −*i*ε*k*η*k*, and hence

$$-i\alpha\_0(\eta\_k^\*\delta(\eta\_k)) = -\alpha\_0(\eta\_k^\*\eta\_k).\tag{10.152}$$

The condition −*i*ω0(*a*∗δ(*a*)) ≥ 0, i.e., eq. (9.56) from Proposition (9.20), therefore implies that ω0(η∗ *k* η*k*) ≤ 0, and hence ω0(η∗ *k* η*k*) = 0 by positivity of ω0. Since *F*0(*H*) is finite-dimensional and *A* ∼= *B*(*F*0(*H*)), cf. (10.98), we may assume ground state(s) to be pure and normal, i.e., there is some unit vector ψ0 ∈ *F*−(*H*) with ω(*a*) = ψ0,*a*ψ0 for each *a* ∈ *A*. Hence ψ0,η∗ *k* η*k*ψ0 = 0, which enforces

$$
\eta\_k \psi\_0 = 0 \ (k = 1, \ldots, N). \tag{10.153}
$$

This property makes ψ0 unique up to a phase. Indeed, together with (10.148) - (10.149), eq. (10.153) implies the values of all one- and two-point functions, i.e.,

$$a\_0(\eta(f)) = a\_0(\eta^\*(f)) = 0;\tag{10.154}$$

$$a\_0(\eta^\*(f)\eta(\mathbf{g})) = a\_0(\eta^\*(f)\eta^\*(\mathbf{g})) = a\_0(\eta(f)\eta(\mathbf{g})) = 0; \quad (10.155)$$

$$
\mathfrak{a}\_{\mathbf{0}}(\mathfrak{\eta}(f)\mathfrak{\eta}^\*(\mathbf{g})) = \langle f, \mathfrak{g} \rangle\_{\mathcal{H}}.\tag{10.156}
$$

Furthermore, the value of ω0 on any product of an odd number of η(*f*) and η∗(*g*) vanishes; for an even number the value ω0(∏*n i*=1 η(*fi*)∏*n j*=1 η∗(*gj*)) it is given by

$$\sum\_{p=1}^{n}(-1)^{n-p}a\mathbf{o}\_{0}(\eta(f\_{1})\eta^{\*}(g\_{p})a\mathbf{o}\_{0}\left(\prod\_{i=2}^{n}\eta(f\_{i})\prod\_{j=1,j\neq p}^{n}\eta^{\*}(g\_{j})\right).$$

Hence (10.153) gives ω0 on all of CAR(C*N*). Since CAR(*H*) = *B*(*F*−(*H*)), this fixes ψ0 up to a phase. Eq. (10.150) is just a fancy way of rewriting (10.153). -

By construction, the ground state energy of (10.147) is zero. In connection with our approach to SSB via Butterfield's Principle it is of interest to compute the energy ε1 of the first excited state. This may be done from (10.120) - (10.121) with (10.122) and the specific expression (10.118) for the quantum Ising chain. Thus we solve

$$\mathcal{A}\,\Psi\_k(\mathbf{x}) + \Psi\_k(\mathbf{x}+1) = \varepsilon\_k \Phi\_k(\mathbf{x}) \; (\mathbf{x} = 1, \dots, N, \Psi\_k(N+1) = 0); \quad (10.157)$$

$$
\lambda \, \varphi\_k(\mathbf{x}) + \varphi\_k(\mathbf{x} - 1) = \varepsilon\_k \, \varphi\_k(\mathbf{x}) \; (\mathbf{x} = 1, \dots, N, \varphi\_k(\mathbf{0}) = \mathbf{0}).\tag{10.158}
$$

A solution of this system (with real wave-functions and positive energy) is given by

$$\mathfrak{g}\_k(\mathbf{x}) = \mathbf{C}(-1)^k \sin(q\_k(\mathbf{x} - N - 1));\tag{10.159}$$

$$
\Psi\_k(\mathbf{x}) = -C \sin(q\_k \mathbf{x});\tag{10.160}
$$

$$\mathfrak{e}\_{k} = \sqrt{1 + \mathcal{X}^{2} + 2\mathcal{X}\cos(q\_{k})},\tag{10.161}$$

where *C* > 0 is a normalization constant, and *qk* should be solved from

$$(N+1)q\_k = (k-1)\pi + \arctan\left(\frac{\sin q\_k}{\lambda + \cos q\_k}\right). \tag{10.162}$$

For example, for λ = 0 (i.e. no transverse magnetic field) we obtain *qk* = *k*π/*N*, where *k* = 1,...,*N*. For λ > 1 there is a unique real solution *qk* for each *k*, too, and even as *N* → ∞ there is an energy gap ε*k* > 0 for each *k*. For 0 < λ < 1, however, there is a *complex* solution *q*1 = π +*i*ρ, where ρ ∈ R is a solution to

$$\tanh((N+1)\rho) = \frac{\sinh \rho}{\cosh \rho - \lambda}.\tag{10.163}$$

As *N* → ∞, we find ρ = −ln(λ)−(1−λ2)λ2(*N*−1) . Eq. (10.161) then gives

$$
\mathfrak{E}(q\_1) \approx (1 - \lambda^2) \lambda^N \ (N \to \ast \ast), \tag{10.164}
$$

which, recalling that *E*(1) *N* = ε1 and *E*(0) *N* = 0 and hence Δ*N* = ε1, confirms (10.82).

#### 10.7 Exact solution of the quantum Ising chain: *N* = ∞

The (two-sided) infinite quantum Ising chain is described by the C\*-algebra

$$F = \mathbf{CAR}(\ell^2(\mathbb{Z}));\tag{10.165}$$

one may also consider a one-sided chain, but it lacks translation symmetry. By the construction at the beginning of the previous section, *F* is isomorphic to the infinite tensor product *A* = *M*2(C)∞. We consider *F* to be generated by the operators *c*± *x* (*x* ∈ Z), where *c*− *x* ≡ *cx* and *c*+ *x* ≡ *c*∗ *x* . In this notation, the CAR (10.95) - (10.96) read

$$[c\_{\mathbf{x}}^{\pm}, c\_{\mathbf{y}}^{\mp}]\_{+} = \delta\_{\mathbf{x}\mathbf{y}};\tag{10.166}$$

$$[c\_x^{\\\pm}, c\_y^{\\\pm}]\_+ = 0.\tag{10.167}$$

Although the local Hamiltonians (10.111) do not have a limit as *N* → ∞, as explained in §10.5 they do generate a time-evolution on *F* in the sense of a continuous homomorphism α : R → Aut(*F*) via (10.65) and (10.67); see also Theorem 9.15.

Let us first extend the approach in the previous section to *N* = ∞, in which case C*N* is replaced by *H* = -2(Z), assuming the theory has already been brought into fermionic form with local Hamiltonians (10.111) (as we will see, it is this step, i.e., the Jordan–Wigner transformation, that marks the difference between *N* < ∞ and *N* = ∞). Thus we define operators *A* : -2(Z) → -2(Z) and *B* : -2(Z) → -2(Z) as the obvious extensions of the *N* × *N* matrices *A* and *B* to operators on -2(Z), and similarly *S* : -2(Z) → -2(Z) is the "full" shift operator, defined by (*S f*)(*x*) = *f*(*x*+1). Instead of the somewhat clumsy explicit solution procedure sketched in the previous section for *N* < ∞, we may now simply rely on the Fourier transformation

$$\mathcal{A}: \ell^2(\mathbb{Z}) \to L^2([-\pi, \pi]);\tag{10.168}$$

$$((\mathcal{F}f)(k) \equiv \hat{f}(k) \;=\sum\_{\mathbf{x} \in \mathbb{Z}} e^{-ik\mathbf{x}} f\_{\mathbf{j}};\tag{10.169}$$

$$(\mathcal{J}^{-1}\hat{f})(\mathbf{x}) \equiv f(\mathbf{x}) = \int\_{-\pi}^{\pi} \frac{dk}{2\pi} e^{ik\mathbf{x}} \hat{f}(k),\tag{10.170}$$

which diagonalizes *A* and *B* to operators *A*ˆ,*B*ˆ : *L*2([−π,π]) → *L*2([−π,π]). For the quantum Ising Hamiltonian (10.110) these are given by the multiplication operators

$$
\hat{A}\Psi(k) = -(\cos k + \lambda)\Psi(k);\tag{10.171}
$$

$$
\mathcal{B}\Psi(k) = -i\sin k \,\Psi(k). \tag{10.172}
$$

For fixed *k*, the eigenvalues and eigenvectors of the 2×2 matrix

$$M\_k = \begin{pmatrix} -(\cos k + \lambda) & -i \sin k \\ i \sin k & \cos k + \lambda \end{pmatrix},\tag{10.173}$$

are ±ε*k*, given by (10.161) with *qk k*. It is then routine to find a unitary 2×2 matrix *Uk* = *uk vk vk uk* that diagonalizes *Mk* in the sense that *U*−1 *k MkUk* = −ε*k* 0 0 ε*k* . Fourier transforming these multiplication operators back to -2(Z) then yields an operator *U* on -2(Z)⊕-2(Z) that satisfies (10.129). This yields a unique ground state ω0 characterized by a property like (10.150) or (10.153), where

$$\eta(\hat{f}) = \int\_{-\pi}^{\pi} \frac{dk}{2\pi} \hat{f}(k)(\mu\_k \hat{c}\_k + \nu\_k \hat{c}\_{-k}^\*);\tag{10.174}$$

$$
\hat{c}\_k = \sum\_{j \in \mathbb{Z}} e^{-ijk} c\_j;\tag{10.175}
$$

$$
\hat{c}\_k^\* = \sum\_{j \in \mathbb{Z}} e^{ijk} c\_j^\*. \tag{10.176}
$$

In summary, one-dimensional fermionic models with quadratic Hamiltonians like (10.111) have a unique ground state even at *N* = ∞. Thus one wonders where SSB in the quantum Ising chain could possibly come from. We will answer this question.

Almost every argument to follow relies on Z2-symmetry. In general, a Z2-action on a C\*-algebra *A* corresponds to an automorphism θ : *A* → *A* such that θ2 = id*A*, i.e. θ represents the nontrivial element of Z2. For example, define θ : *F* → *F* by

$$\theta(c\_{\chi}^{\pm}) = -c\_{\chi}^{\pm} \ (j \in \mathbb{Z}),\tag{10.177}$$

which is an example of a Bogoliubov transformation (cf. Proposition 10.12) and hence extends to an automorphism of *F* (which implies that θ(1*F*) = 1*F*). Clearly, θ2 = id*F*, and in addition each local Hamiltonian (10.111) is invariant under θ; by implication, so is the dynamics α, i.e., α*t* ◦ θ = θ ◦α*t* for all *t* ∈ R.

A C\*-algebra *A* carrying a Z2-action decomposes as

$$A = A\_{+} \oplus A\_{-} ; \tag{10.178}$$

$$A\_{\pm} = \{ a \in A \mid \theta(a) = \pm a \},\tag{10.179}$$

where the *even* part *A*+ is a subalgebra of *A*, whereas the *odd* part *A*− is not: one has *ab* ∈ *A*+ for *a*,*b* both in either *A*+ or *A*−, and *ab* ∈ *A*− if one is in *A*+ and the other in *A*−. For example, if *A* = *B*(*H*) for some Hilbert space *H* and *w* : *H* → *H* is a untitary operator satisfying *w*2 = 1 (and hence *w*∗ = *w*), then

$$\theta(a) = \bowtie{a} ww^\* \left(= \bowtie{a} ww\right) \tag{10.180}$$

defines a Z2-action on *A*. In that case, *A*+ and *A*− consist of all *a* ∈ *A* that commute and anticommute with *w*, respectively, that is,

$$A\_{\pm} = \{ a \in A \mid aw \mp wa = 0 \}. \tag{10.181}$$

In case of (10.165) with (10.177), the subspace *F*+ (*F*−) is just the linear span of all products of an even (odd) number of *c*± *j* 's.

Let us move to Theorem C.90 and reconsider the proof of the claim that if πω(*A*) = C · 1, then ω is mixed. If the commutant πω(*A*) is nontrivial, then it contains a nontrivial projection *e*+ ∈ πω(*A*) . It then follows that *e*+Ωω = 0: for if *e*+Ωω = 0, then *ae*+Ωω = *e*+*a*Ωω = 0 for all *a* ∈ *A*, so that *e*+ = 0, since πω is cyclic. Similarly, *e*−Ωω = 0 with *e*− = 1*H* −*e*+, so we may define the unit vectors

$$\mathfrak{Q}\_{\pm} = e\_{\pm} \mathfrak{Q}\_{\mathfrak{O}} / \left| \left| e\_{\pm} \mathfrak{Q}\_{\mathfrak{O}} \right| \right|, \tag{10.182}$$

and the associated states ω±(*a*) = Ω±,πω(*a*)Ω±on *A*. This yields a convex decomposition ω = λω+ + (1−λ)ω−, with λ = Ω−2. Since λ = 0,1 and ω+ = ω−, it follows that ω is mixed. The associated reduction is effected by writing

$$H = H\_+ \oplus H\_-;\tag{10.183}$$

$$H\_{\pm} = e\_{\pm}H,\tag{10.184}$$

in that *A* (more precisely, πω(*A*)) maps each subspace *H*± into itself. Now pass from the projections *e*± to the operator *w* = *e*+ −*e*−, which by construction satisfies

$$
\omega^\* = \omega^{-1} = \text{w.}\tag{10.185}
$$

In particular, *w* is unitary. Conversely, if some unitary *w* satisfies *w*2 = 1*H*, then

$$e\_{\pm} = \frac{1}{2} (1\_H \pm w) \tag{10.186}$$

are projections satisfying *e*+ +*e*− = 1*H*, giving rise to the decomposition (10.184). Group-theoretically, this means that one has a unitary Z2-action on *H* ≡ *H*ω, in which the nontrivial element of Z2 = {−1,1} is represented by *w*. The decomposition (10.184) then simply means that Z2 acts trivially on *H*+ (in that both group elements are represented by the unit operator) and acts nontrivially on *H*− (in that the nontrivial element is represented by *minus* the unit operator). In conclusion, one has a Z2 perspective on the reduction of *H*ω, and instead of a projection *e* ∈ πω(*A*) one may equivalently look for an operator *w* ∈ πω(*A*) that satisfies (10.185).

Proposition 10.14. *Suppose A carries a* Z2*-action* θ *and consider a state* ω : *A* → C *that is* Z2*-invariant in the sense that* ω(θ(*a*)) = ω(*a*) *for all a* ∈ *A. We write this as* θ∗ω = ω*, with* θ∗ω = ω ◦ θ*. Then there is a unitary operator w* : *H*ω → *H*ω *satisfying w*2 = 1*H, w*Ω = Ω*, and and w*πω(*a*)*w*∗ = πω(θ(*a*)) *for each a* ∈ *A.*

Cf. Corollary 9.12. In this situation, we obtain a decomposition of *H* ≡ *H*ω according to (10.183), where the projections *e*± are given by (10.186), so that, equivalently,

$$H\_{\pm} = \{ \Psi \in H \mid \nu \Psi = \pm \Psi \} = A\_{\pm} \Omega\_{-} \,. \tag{10.187}$$

In terms of the decomposition (10.178), it is easily seen that each subspace *H*± is stable under *A*+, whereas *A*− maps *H*± into *H*∓. We denote the restriction of πω(*A*+) to *H*± by π±, so that a Z2-invariant state θ on *A* not just gives rise to the GNS-representation πω of *A* on *H*ω, but also induces two representations π± of the even part *A*+ on *H*±. This leads to a refinement of Theorem C.90:

Theorem 10.15. *Suppose A carries a* Z2*-action* θ*, and let* ω : *A* → C *be a* Z2 *invariant state. With the above notation, suppose the representation* π+(*A*+) *on H*+ *is irreducible. Then also the representation* π−(*A*+) *on H*− *is irreducible, and there are the following two possibilities for the representation* πω(*A*) *on H* = *H*+ ⊕*H*−*:*

*1.* πω(*A*) *is irreducible (and* ω *is pure) iff* π+(*A*+) *and* π−(*A*+) *are inequivalent. 2.* πω(*A*) *is reducible (and* ω *is mixed) iff* π+(*A*+) *and* π−(*A*+) *are equivalent.*

*Proof.* The proof of this theorem is much more difficult than one would expect (given its simple statement), so we restrict ourselves to the easy steps, as well as to two examples illustrating each of the two possibilities. To start with the latter:

1. *A* = *M*2(C), with θ(*a*) = σ3*a*σ3; note that σ2 3 = 1 and σ∗ 3 = σ3. Then

$$A\_+ = \left\{ \begin{pmatrix} z\_+ & 0 \\ 0 & z\_- \end{pmatrix}, z\_\pm \in \mathbb{C} \right\} \equiv D\_2(\mathbb{C});\tag{10.188}$$

$$A\_{-}=\left\{ \begin{pmatrix} 0 \ z\_{1} \\ z\_{2} \ 0 \end{pmatrix}, z\_{1}, z\_{2} \in \mathbb{C} \right\},\tag{10.189}$$

where *Dn*(C) denotes the C\*-algebra of diagonal *n*×*n* matrices. Take Ω = (1,0), with associated state

$$a\mathfrak{o}(a) = \langle \mathfrak{Q}, a\mathfrak{Q} \rangle,\tag{10.190}$$

where *a* ∈ *M*2(C). It follows from §2.4 that the associated GNS-representation πω(*A*) is just (equivalent to) the defining representation of *M*2(C) on *H*ω = C2, in which the cyclic vector Ωω of the GNS-construction is Ω itself. Since σ3Ω = Ω, the state defined by (10.190) is Z2-invariant, and the unitary operator *w* in Proposition 10.14 is simply *w* = σ3. Hence the decomposition (10.183) of *H* = C2 is simply C2 = C⊕C, i.e.,

$$H\_+ = \{(z, 0), z \in \mathbb{C}\};\tag{10.191}$$

$$H\_- = \{(0, z), z \in \mathbb{C}\}.\tag{10.192}$$

Of course, we then have *H*± = *A*±Ω. Identifying *H*± ∼= C, this gives the onedimensional representations π±(*D*2(C)) as

$$
\pi\_{\pm} \begin{pmatrix} z\_{+} & 0 \\ 0 & z\_{-} \end{pmatrix} = z\_{\pm}, \tag{10.193}
$$

which are trivially inequivalent. Hence by Theorem 10.15 the defining representation of *M*2(C) on C2 is irreducible, as it should be.

2. *A* = *D*2(C), with

$$\theta(\text{diag}(z\_+, z\_-)) = \text{diag}(z\_-, z\_+), \tag{10.194}$$

where we have denoted the matrix in (10.188) by diag(*z*+,*z*−). This time,

$$A\_{\pm} = \{ \text{diag}(z, \pm z), z \in \mathbb{C} \}. \tag{10.195}$$

We once again define a Z2-invariant state ω by (10.190), but this time we take

10.7 Exact solution of the quantum Ising chain: *N* = ∞ 401

$$
\mathfrak{Q} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix}. \tag{10.196}
$$

Hence

$$H\_{\pm} = \{ (z, \pm z), z \in \mathbb{C} \} \dots \tag{10.197}$$

We may now identify each *A*± with C under the map diag(*z*,±*z*) → *z* from *A*± to C. Similarly, we identify each each subspace *H*± with C under the map *H*± → C defined by (*z*,±*z*) → *z*. Under these identifications, we have two onedimensional representations π± of the C\*-algebra C on the Hilbert space C, given by π±(*z*) = *z*. Clearly, these are equivalent: they are even identical. Hence by Theorem 10.15 the defining representation of *D*2(C) on C2 is reducible, as it should be: the explicit decomposition of C2 in *D*2(C)-invariant subspaces is just the one (10.191) - (10.192) of the previous example.

The first-numbered claim of Theorem 10.15 is relatively easy to prove from Theorem C.90. Suppose π±(*A*+) are inequivalent and take *b* ∈ πω(*A*) : we want to show that *b* = λ · 1 for some λ ∈ C. Relative to *H* = *H*+ ⊕*H*−, we write

$$b = \begin{pmatrix} b\_{++} \ b\_{+-} \\ b\_{-+} \ b\_{--} \end{pmatrix},\tag{10.198}$$

where the four operators in this matrix act as follows:

$$b\_{++}: H\_+ \to H\_+, b\_{+-}: H\_- \to H\_+, b\_{-+}: H\_+ \to H\_-, b\_{--}: H\_- \to H\_-. \tag{10.199}$$

Since *A*+ ⊂ *A*, we also have *b* ∈ πω(*A*+) . The condition [*b*,*a*] = 0 for each *a* ∈ *A*+ is equivalent to the four conditions

$$[b\_{++}, \pi\_+(a)] = 0;\tag{10.200}$$

$$[b\_{--}, \pi\_{-}(a)] = 0;\tag{10.201}$$

$$
\pi\_+(a)b\_{+-} = b\_{+-}\pi\_-(a);\tag{10.202}
$$

$$
\pi\_-(a)b\_{-+} = b\_{-+}\pi\_+(a). \tag{10.203}
$$

We now use the fact (which we state without proof) that, as in group theory, the irreducibility and inequivalence of π±(*A*+) implies that there can be no nonzero operator *c* : *H*+ → *H*− such that *c*π+(*a*) = π−(*a*)*c* for all *a* ∈ *A*+, and vice versa. Hence *b*+− = 0 as well as *b*−+ = 0. In addition, the irreducibility of π±(*A*+) implies that *b*++ = λ+ · 1*H*+ and *b*−− = λ− · 1*H*1 . Finally, the property [*b*,*a*] = 0 for each *a* ∈ *A*− implies λ+ = λ−. Hence *b* = λ · 1, and πω(*A*) is irreducible.

To prove the second-numbered claim of Theorem 10.15, let π+(*A*+) ∼= π−(*A*+), so by definition (of equivalence) there is a unitary operator *v* : *H*− → *H*+ such that

$$\text{v } \mathfrak{m}\_{-}(a) = \mathfrak{m}\_{+}(a)\mathfrak{v}, \forall a \in A\_{+}. \tag{10.204}$$

Extend *v* to an operator *w* : *H* → *H* by

$$
\hat{\mathbf{v}} \cdot \mathbf{w} = \begin{pmatrix} \mathbf{0} & \nu \\ \nu^\* \ \mathbf{0} \end{pmatrix}. \tag{10.205}
$$

It is easy to verify from (10.204) that [*w*,π(*a*)] = 0 for each *a* ∈ *A*+. To check that the same is true for each *a* ∈ *A*−, one needs the difficult analytical fact that *w* is a (weak) limit of operators of the kind π(*an*), where *an* ∈ *A*−, which also implies that *w*∗π(*a*) ∈ π(*A*+). Since π(*A*+) = π(*A*+) and *w* ∈ π(*A*+) , we obtain [*w*∗π(*a*),*w*] = 0 for each *a* ∈ *A*−. But for unitary operators *w* this is the same as [*w*,π(*a*)] = 0. So *w* ∈ π(*A*) , and hence π(*A*) is reducible by Theorem C.90. -

In determining the ground state(s) of the quantum Ising chain, we will apply Theorem 10.15 to the C\*-algebra (10.87). This application relies on the representation theory of *F*. For the moment we leave the Hilbert space *H* general, equipped though with a conjugation *J* : *H* → *H*. It turns out to be convenient to use the *self-dual formulation of the CAR*, which treats *c* and *c*∗ on an equal footing. Define

$$K = H \oplus H,\tag{10.206}$$

whose elements are written as *h* = (*f*,*g*) or *h* = *f*+˙ *g*, with inner product

$$
\langle h\_1, h\_2 \rangle\_K = \langle f\_1, f\_2 \rangle\_H + \langle \mathbf{g}\_1, \mathbf{g}\_2 \rangle\_H. \tag{10.207}
$$

We then introduce a new operator in CAR(*H*), namely the *field*

$$\Phi(h) = c^\*(f) + c(Jg),\tag{10.208}$$

which is *linear* in *h* = *f*+˙ *g*, because the antilinearity of *c*(*f*) in *f* is canceled by the antilinearity of *J*. This yields the anti-commutation relations

$$[\Phi^\*(h\_1), \Phi(h\_2)]\_+ = \langle h\_1, h\_2 \rangle\_K,\tag{10.209}$$

but be aware that generally [Φ∗(*h*1),Φ∗(*h*2)]+ and [Φ(*h*1),Φ(*h*2)]+ do not vanish. Indeed, in terms of the antilinear operator Γ : *K* → *K*, defined by

$$
\Gamma = \begin{pmatrix} 0 \ J \\ J \ 0 \end{pmatrix} \tag{10.210}
$$

we have the following expression for the adjoint Φ(*h*)∗ ≡ Φ∗(*h*):

$$
\Phi^\*(h) = \Phi(\Gamma h). \tag{10.211}
$$

If we identify *f* ∈ *H* with *f*+˙ 0 ∈ *K*, we may reconstruct *c* and *c*∗ from Φ through

$$c^\*(f) = \Phi(f);\tag{10.212}$$

$$c(f) = \Phi(\Gamma f). \tag{10.213}$$

Bogoliubov transformations now take an extremely elegant form. For any unitary operator *S* on *K* that satisfies [*S*,Γ ] = 0, we define the transform Φ*S* of Φ by

10.7 Exact solution of the quantum Ising chain: *N* = ∞ 403

$$\Phi\_{\mathbb{S}}(h) = \Phi(Sh),\tag{10.214}$$

with associated creation- and annihilation operators (where *H f* ≡ *f*+˙ 0, as above)

$$c^\*\_{\mathcal{S}}(f) = \Phi\_{\mathcal{S}}(f);\tag{10.215}$$

$$c\_{\mathcal{S}}(f) = \Phi\_{\mathcal{S}}^\*(f). \tag{10.216}$$

To see the equivalence with the original formulation of the Bogoliubov transformation, note that for unitary *S*, the condition [*S*,Γ ] = 0 is equivalent to the structure

$$S = \begin{pmatrix} \mu & \nu J \\ J\nu \, J\mu J \end{pmatrix},\tag{10.217}$$

where *u* : *H* → *H* is linear, *v* : *H* → *H* is antilinear, and *u* and *v* satisfy (10.133) - (10.134). Moreover, from (10.137) - (10.138) we obtain

$$c\_S(f) = \mathfrak{n}(f);\tag{10.218}$$

$$c\_S^\*(f) = \eta^\*(f). \tag{10.219}$$

An interesting class of pure states on CAR(*H*) arises as follows.

Theorem 10.16. *There is a bijective correspondence between:*

• *Projections e* : *K* → *K that (apart form the properties e*2 = *e*∗ = *e) satisfy*

$$
\Gamma e\Gamma = \mathbf{1}\_K - e;\tag{10.220}
$$

• *States* ω*e on F that satisfy*

$$\alpha\_e(\Phi(h)^\*\Phi(h)) = \langle h, eh \rangle \,\forall h \in K. \tag{10.221}$$

*Such a state* ω*e is automatically pure (so that the corresponding* GNS*-representation* π*e is irreducible), and is explicitly given by*

$$
\mathfrak{a}\_{\mathfrak{e}}(\Phi(h\_1)\cdots\Phi(h\_{2n+1})) = 0;\tag{10.222}
$$

$$\mathfrak{a}\_{\epsilon}(\Phi(h\_1)\cdots\Phi(h\_{2n})) = \sum\_{p \in \mathfrak{S}\_{2n}}' \text{sgn}(p) \prod\_{j=1}^{n} \langle eh\_{\text{sgn}(2j)}, \Gamma h\_{\text{sgn}(2j-1)} \rangle (10.223)$$

*the sum* Σ *is over all permutations p of* 1,...,2*n such that*

$$p(2j-1) < p(2j);\tag{10.224}$$

$$p(1) < p(\mathfrak{z}) < \dots < p(2n-1). \tag{10.225}$$

We omit the proof. Note that (10.221) is a special case of (10.223), because of (10.211). States like ω*e*, which are determined by their two-point functions, are called *quasi-free*; the ground state ω0 on CAR(C*N*) constructed in the previous section is an example (one also has mixed quasi-free states, e.g. certain KMS states).

As a warm-up, we reconstruct the ground state of the free fermionic Hamiltonian on *F* using the above formalism. That is, we assume that *hN* in (10.111) reads

$$h\_N = \sum\_{\mathbf{x}=-N/2}^{N/2-1} \mathfrak{E}\_{\mathbf{x}} \mathfrak{c}\_{\mathbf{x}}^\* \mathfrak{c}\_{\mathbf{x}},\tag{10.226}$$

initially defining dynamics on *FN* = CAR(C*N*). In that case, the projection *e*0 onto the second copy of *H* = C*N* in *K*, i.e.

$$e\_0 = \begin{pmatrix} 0 \ 0 \\ 0 \ 1 \end{pmatrix},\tag{10.227}$$

reproduces the ground state ω0(*a*) = 0|*a*|0, where |0 is the vector 1 ∈ C in *F*−(*H*), such that *c*(*f*)|0 = 0 for all *f* ∈ *H*. This also works for *N* = ∞, i.e., we construct dynamics on CAR(-2(Z)) from the local Hamiltonians (10.226) as indicated at the beginning of this section, and use the same formula for *e*0, this time with *H* = -2(Z).

In the more general case (10.111), we replace *e*0 in (10.227) by

$$e\_0^{(S)} = S e\_0 S^{-1},\tag{10.228}$$

where *S* is given by (10.217), in which for *N* < ∞ the operators *u* and *v* were constructed in (10.131) - (10.132). This time, the associated state ω*e* (*S*) 0 ≡ ω*S* is the state called ω0 in Theorem 10.13. As explained at the beginning of this section, this procedure even works for *N* = ∞ and hence *H* = -2(Z).

Having understood fermionic models with quadratic Hamiltonians, what remains to be done now is to reformulate the original quantum Ising chain, defined in terms of the local spin matrices σ*i*(*x*), in terms of the fermionic variables *cx* and *c*∗ *x* . For finite *N* this was done through the Jordan–Wigner transformation (10.102) - (10.103). This time we need a similar isomorphism between *A* and *F*, where

$$A = \otimes\_{j \in \mathbb{Z}} \mathcal{M}\_2(\mathbb{C});\tag{10.229}$$

$$F = \mathbf{CAR}(\ell^2(\mathbb{Z})),\tag{10.230}$$

and hence we would need to start the sums in the right-hand side of (10.102) - (10.103) at *j* = −∞. At first sight this appears to be impossible, though, because operators like exp(π*i*∑*x*−1 *y*=−∞ σ+(*y*)σ−(*y*)) do not lie in *A* (whose elements have infinite tails of 2×2 unit matrices). Fortunately, this problem can be solved by adding a formal operator *T* to *A*, which plays the role of the "tail"

$$\mathbf{u}^{\star\star}T = e^{\pi i \sum\_{\gamma=-\omega}^{0} \sigma\_{+}(\mathbf{y})\sigma\_{-}(\mathbf{y})) \mathbf{v}\_{.} \tag{10.231}$$

This formal expression (to be used only heuristically) suggests the relations:

10.7 Exact solution of the quantum Ising chain: *N* = ∞ 405

$$T^2 = 1;\tag{10.232}$$

$$T^\* = T;\tag{10.233}$$

$$T a T = \theta\_{-}(a),\tag{10.234}$$

where θ− : *A* → *A* is a Z2-action defined by (algebraic) extension of

$$\theta\_{-}(\sigma\_{\pm}(\mathbf{y})) = -\sigma\_{\pm}(\mathbf{y})\ (\mathbf{y} \le \mathbf{0});\tag{10.235}$$

$$
\theta\_{-}(\mathfrak{σ}\_{\pm}(\mathfrak{y})) = \mathfrak{σ}\_{\pm}(\mathfrak{y}) \text{ ( $\mathfrak{y} > 0$ );}\tag{10.236}
$$

$$\theta\_{-}(\sigma\_{\mathbb{S}}(\mathbf{y})) = \sigma\_{\mathbb{S}}(\mathbf{y}) \text{ ( $\mathbf{y} \in \mathbb{Z}$ );}\tag{10.237}$$

$$\theta\_{-}(\sigma\_{0}(\mathbf{y}) = \sigma\_{0}(\mathbf{y}) \text{ ( $\mathbf{y} \in \mathbb{Z}$ )},\tag{10.238}$$

where σ0 = 12. Formally, define an algebra extension

$$
\hat{A} = A \oplus A \cdot T,\tag{10.239}
$$

with elements of the type *a*+*bT*, *a*,*b* ∈ *A*, and algebraic relations given by (10.232) - (10.233). That is, we have

$$(a+bT)^{\*} = a^{\*} + \theta\_{-}(b^{\*})T;\tag{10.240}$$

$$(a+bT)\cdot(a'+b'T) = aa'+b\theta\_-(b')+(ab'+b\theta\_-(a'))T.\qquad(10.241)$$

Within *A*ˆ, the correct version of (10.102) - (10.103) may now be written down as

$$c\_{\boldsymbol{x}}^{\pm} = T e^{\mp \pi i \sum\_{\gamma=\times}^{0} \sigma\_{+}(\boldsymbol{y}) \sigma\_{-}(\boldsymbol{y})} \sigma\_{\boldsymbol{x}}^{\pm} \; (\boldsymbol{x} < 1);\tag{10.242}$$

$$c\_\chi^\pm = T\sigma\_1^\pm;\tag{10.243}$$

$$c\_{\chi}^{\pm} = T e^{\mp \pi i \Sigma\_{\gamma - 1}^{\chi - 1} \sigma\_{+}(\chi) \sigma\_{-}(\chi)} \sigma\_{\chi}^{\pm} \ (\times > 1),\tag{10.244}$$

with formal inverse transformation given by

$$\sigma\_{\pm}(\mathbf{x}) = T e^{\pm \pi i \sum\_{\mathbf{y}=\mathbf{x}}^{0} c\_{\mathbf{y}}^{+} c\_{\mathbf{y}}^{-}} c\_{\mathbf{x}}^{\pm} \ (\mathbf{x} < 1);\tag{10.245}$$

$$
\sigma\_{\pm}(\mathbf{x}) = T c\_1^{\pm};\tag{10.246}
$$

$$\sigma\_{\pm}(\mathbf{x}) = T e^{\pm \pi l \sum\_{\mathbf{y}=1}^{\chi-1} \sigma\_{+}(\mathbf{y}) \sigma\_{-}(\mathbf{y})} \sigma\_{\pm}(\mathbf{x}) \text{ ( $\mathbf{x} > 1$ )},\tag{10.247}$$

where this time we regard *T* as an element of the extended fermionic algebra

$$
\hat{F} = F \oplus F \cdot T,\tag{10.248}
$$

satisfying the same rules (10.232) - (10.234), but now in terms of a "fermionic" Z2 action θ*y* : *F* → *F* given by extending the following action on elementary operators:

$$\theta\_{-}(c\_{\circ}^{\pm}) = -c\_{\circ}^{\pm} \text{ ( $\chi \le 0$ );}\tag{10.249}$$

$$
\theta\_{-}(c\_{\circ}^{\pm}) = c\_{\circ}^{\pm}(\mathbf{y} > \mathbf{0}).\tag{10.250}
$$

(10.251)

Because of *T*, the Jordan–Wigner transformation does not give an isomorphism *A* ∼= *F*, but it does give an isomorphism *A*ˆ ∼= *F*ˆ. More importantly, if, having already defined the Z2-action θ on *F* by (10.177), we define a similar Z2-action on *A* by

$$\theta(\mathfrak{o}\_{\pm}(\mathfrak{y})) = -\mathfrak{o}\_{\pm}(\mathfrak{y}) \text{ ( $\mathfrak{y} \in \mathbb{Z}$ )};\tag{10.252}$$

$$\theta(\mathfrak{o}\_3(\mathfrak{y})) = \mathfrak{o}\_3(\mathfrak{y}) \text{ ( $\mathfrak{y} \in \mathbb{Z}$ )};\tag{10.253}$$

$$\theta(\mathfrak{o}\_0(\mathfrak{y})) = \mathfrak{o}\_0(\mathfrak{y}) \text{ ( $\mathfrak{y} \in \mathbb{Z}$ )},\tag{10.254}$$

and decompose *A* = *A*+ ⊕*A*− and *F* = *F*+ ⊕*F*−, according to this action, cf. (10.178), we have isomorphisms

$$A\_{+} \cong F\_{+};\tag{10.255}$$

$$A\_{-} \cong F\_{-}T;\tag{10.256}$$

$$A \cong F\_+ \oplus F\_- T. \tag{10.257}$$

For given dynamics (10.111), suppose ω*A* 0 is a Z2*-invariant* ground state on *A*. Then ω*A* 0 also defines a Z2-invariant ground state ω*F* 0 on *F* by (10.255) and ω*F* 0 (*f*) = 0 for all *f* ∈ *F*−. Conversely, a Z2-invariant ground state ω*F* 0 on *F* defines a state ω*A* 0 on *A* by (10.255) and ω*A* 0 (*a*) = 0 for all *a* ∈ *A*−. But *F* has a unique ground state, so:

Theorem 10.15 gives a representation-theoretical criterion deciding between these possibilities, but to apply it we need some information on the restriction of Z2 invariant quasi-free pure states on *F* to its even part *F*+. The abstract setting involves a Z2-action *W* on *K* that commutes with Γ (so that *W* is unitary, *W*2 = 1, and [Γ ,*W*] = 0), which induces a Z2-action θ on *F* by linear and algebraic extension of θ(Φ(*h*)) = Φ(*W h*). A quasi-free state ω*e*, defined according to Theorem 10.16 by a projection *e* : *K* → *K* that satisfies (10.220), is then Z2-invariant iff [*W*, *e*] = 0.

In our case, this simplifies to θ(Φ(*h*)) = −Φ(*h*), so that *W* = −1, and every projection commutes with *W*. In any case, with considerable effort one can prove:

Lemma 10.17. *Given some* Z2*-action W on K, as well as a projection e* : *K* → *K satisfying* (10.220)*, such that* [*W*,Γ ]=[*W*, *e*] = 0*:*

Theorem 10.15 then leads to a lemma, which also summarizes the discussion so far.

Lemma 10.18. *1. For given* Z2*-invariant dynamics, let* ω*F* 0 *be the (unique,* Z2 *invariant) ground state on F* = *F*+ ⊕ *F*−*. Under F*+ ⊂ *F the associated* GNS*representation space H*0 *decomposes as H*0 = *H*+ 0 ⊕*H*− 0 *, with H*± 0 = *F*±Ω0*, and we denote the restriction of* π0(*F*+) *to H*± 0 *by* π± 0 *. Then* π± 0 (*F*+) *are irreducible.*

	- *a. Then* ω*A* 0 *is a ground state on A. Any* Z2*-invariant ground state on A arises in this way (via F ), so that there is a unique* Z2*-invariant ground state on A.*
	- *b. The state* ω*A* 0 *is pure on A iff the irreducible representations* π*T* +(*F*+) *(or* π+ 0 (*F*+)*) and* π*T* −(*F*+) *are inequivalent.*

It turns out to be difficult to directly check the (in)equivalence of π*T* ±(*F*+). Fortunately, we can circumvent this problem by passing to yet another (irreducible) representation of *F*+. We first enlarge *F* to a new algebra

$$
\hat{F} = F \oplus FT = F\_+ \oplus F\_- \oplus F\_+ T \oplus F\_- T,\tag{10.258}
$$

and extend the state ω*F* 0 on *F* to a state ωˆ0 on *F*ˆ by putting ωˆ0(*FT*) = 0, so that ωˆ0 is nonzero only on *F*+ ⊂ *F*ˆ. Let πˆ0 be the associated GNS-representation of *F*ˆ on the Hilbert space *H*ˆ0 = *F*ˆΩˆ . Under πˆ(*F*+) this space decomposes as

$$
\hat{H}\_0 = \overline{F\_+\hat{\Omega}\_0} \oplus \overline{F\_-\hat{\Omega}\_0} \oplus \overline{F\_+T\hat{\Omega}\_0} \oplus \overline{F\_-T\hat{\Omega}\_0},\tag{10.259}
$$

with corresponding restrictions πˆ±(*F*+) and πˆ*T* ±(*F*+); more precisely, πˆ± is the restriction of πˆ(*F*+) to *F*±Ωˆ 0, whilst πˆ*T* ± is is the restriction of πˆ(*F*+) to *F*±*T*Ωˆ 0. Clearly, πˆ±(*F*+) is the same as π± 0 (*F*+), and πˆ*T* −(*F*+) is just our earlier π*T* −(*F*+), but πˆ*T* +(*F*+) is new. To understand the latter, we rewrite (10.259) as

$$
\hat{H}\_0 = H\_0 \oplus \hat{H}\_0^T;\tag{10.260}
$$

$$H\_0 = \underline{F\_+\hat{\mathbf{2}}\_0 \oplus F\_-\hat{\mathbf{2}}\_0} \cong \overline{F\_+\underline{\mathbf{2}}\_0} \oplus \overline{F\_-\underline{\mathbf{2}}\_0};\tag{10.261}$$

$$
\hat{H}\_0^T = F\_+ T \hat{\mathfrak{Q}}\_0 \oplus F\_- T \hat{\mathfrak{Q}}\_0,\tag{10.262}
$$

the point being that πˆ(*F*) evidently restricts to both *H*0 and *H*ˆ *T* 0 . We know the action of πˆ(*F*) on *H*0 quite well: it is the representation induced by the ground state ω0. As to *H*ˆ *T* 0 , we define a state ωˆ *T* 0 on *F* by

$$
\langle \hat{\mathfrak{H}}\_0^T(a) = \langle \hat{\mathfrak{H}}(T)\hat{\mathfrak{Q}}\_0, \hat{\mathfrak{H}}(a)\hat{\mathfrak{H}}(T)\hat{\mathfrak{Q}}\_0 \rangle\_{\hat{H}\_0} = \langle \hat{\mathfrak{Q}}\_0, \hat{\mathfrak{H}}(\theta\_-(a))\hat{\mathfrak{Q}}\_0 \rangle\_{\hat{H}\_0},\tag{10.263}
$$

where the second equality follows from (10.234). Comparing *H*0 and *H*ˆ0, for all *b* ∈ *F* (and hence especially for *b* = θ−(*a*)) we simply have

$$
\langle \langle \hat{\Omega}\_0, \hbar(b)\hat{\Omega}\_0 \rangle\_{\hat{H}\_0} = \mathfrak{d}\_0(b) = \mathfrak{a}\_0^F(b), \tag{10.264}
$$

so that ωˆ *T* 0 = ω*F* 0 ◦ θ− ≡ θ∗ −ω*F* 0 . Decomposing the GNS-representation space *H*θ∗ −ω*F* 0 of πθ∗ −ω*F* 0 (*F*) as *H*θ∗ −ω*F* 0 = *H*+ θ∗ −ω*F* 0 ⊕*H*− θ∗ −ω*F* 0 , it follows that πˆ*T* +(*F*+) is the restriction of πθ∗ −ω*F* 0 (*F*+) to *H*+ θ∗ −ω*F* 0 . Therefore, the representation πˆ(*F*) restricted to *H*ˆ *T* 0 is the GNS-representation πθ∗ −ω*F* 0 (*F*), so that in turn πˆ*T* +(*F*+) is πθ∗ −ω*F* 0 (*F*+), restricted to *H*+ θ∗ −ω0 . Hence, further to (10.260) - (10.262), we obtain the decomposition

$$
\hat{\mathfrak{star}}(F) \cong \pi\_{a\_0^F}(F) \oplus \pi\_{\theta\_-^\* a\_0^F}(F). \tag{10.265}
$$

The point is that for the quantum Ising chain Hamiltonian (10.110), we have:

The first claim follows from Theorem 10.20 below. The third follows from Lemma 10.18 and the previous claims. The second claim is proved by repeatedly applying Theorem 10.15 to πˆ(*F*ˆ). Given this lemma, the real issue now lies in comparing πω*F* 0 and πθ∗ −ω*F* 0 , both as representations of *F* (as they are defined) and as representations of *F*+ ⊂ *F*. This can be settled in great generality by first looking at Theorem 10.16, and thence, recalling the positive-energy projection (10.228), realizing that

$$
\pi\_{\alpha\_0^F} = \pi\_{e\_0^{(S)}}; \tag{10.266}
$$

$$
\pi\_{\theta\_-^\* a\_0^F} = \pi\_{W\_- e\_0^{(S)} W\_-}.\tag{10.267}
$$

Here *W*− : *K* → *K* is the Z2-action on *K* defining the Z2-action θ− on *F* as explained above Lemma 10.17; specifically, *W*− is the direct sum of two copies of *w*− : -2(Z) → -2(Z), defined by *w*−(*fj*) = *fj* (*j* > 0) and *w*−(*fj*) = −*fj* (*j* ≤ 0).

Subsequently, without proof we invoke a basic result on the CAR-algebra:

Theorem 10.20. *Let e and e be projections on K that satisfy* (10.220)*. Then:*

$$\begin{array}{l} \text{1. } \mathfrak{m}\_{\epsilon}(F) \cong \mathfrak{m}\_{\epsilon'}(F) \text{ iff } e - e' \in B\_2(K);\\ \text{2. } \mathfrak{m}\_{\epsilon}^{+}(F\_{+}) \cong \mathfrak{m}\_{\epsilon'}^{+}(F\_{+}) \text{ iff } e - e' \in B\_2(K) \text{ and } \text{dim}(eK \cap (1 - e')K) \text{ is even.} \end{array}$$

If the first condition is satisfied, the dimension in the second part is finite, so that one may indeed say it is even or odd. From Lemmas 10.18 and 10.19 and Theorem 10.20, we finally obtain the phase structure of the infinite quantum Ising chain:

Theorem 10.21. *The unique* Z2*-invariant ground state* ω0 *of the Hamiltonian* (10.110) *is pure (and hence forms the unique ground state) iff both of the following hold:*

$$e\_0^{(S)} - W\_- e\_0^{(S)} W\_- \in \mathcal{B}\_2(K);\tag{10.268}$$

$$\dim(e\_0^{(S)}K \cap (1 - W\_- e\_0^{(S)}W\_-)K) \text{ is even.}\tag{10.269}$$

*This is true for all* λ *with* |λ| ≥ 1*. If* |λ| < 1*, then* ω0 = 1 2 (ω+ 0 +ω− 0 )*, where* ω± 0 *are pure and transform under the* Z2*-action* θ *as* ω± 0 ◦ θ = ω∓ 0 *.*

#### 10.8 Spontaneous symmetry breaking in mean-field theories

We are now going to study SSB in so-called *mean-field theories*: these are quantum spin systems with Hamiltonians like the *Curie–Weiss-model* for ferromagnetism:

$$h\_{\Lambda}^{\rm CW} = -\frac{J}{2|\Lambda|} \sum\_{\mathbf{x}, \mathbf{y} \in \Lambda} \sigma\_3(\mathbf{x}) \sigma\_3(\mathbf{y}) - B \sum\_{\mathbf{x} \in \Lambda} \sigma\_1(\mathbf{x}),\tag{10.270}$$

where *J* > 0 scales the spin-spin coupling, and *B* is an external magnetic field. Similar to the quantum Ising model, (10.270) has a Z2-symmetry (σ1,σ2,σ3) 7→ (σ1,−σ2,−σ3), which at each site *x* is implemented by *u*(*x*) = σ1(*x*). This model differs from its short-range counterpart (9.42), i.e, the quantum Ising model, or the Heisenberg model (9.44), in that every spin now interacts with every other spin. It falls into the class of *homogeneous mean-field theories*, which are defined by a single-site Hilbert space *Hx* = *H* = C *n* and local Hamiltonians of the type

$$h\_{\Lambda} = |\Lambda| \tilde{h}(T\_0^{(\Lambda)}, T\_1^{(\Lambda)}, \dots, T\_{n^2 - 1}^{(\Lambda)}). \tag{10.271}$$

Here *T*0 = 1*n*, and the matrices (*Ti*) *n* 2−1 *i*=1 in *Mn*(C) form a basis of the real vector space of traceless self-adjoint *n*×*n* matrices; the latter may be identified with *i* times the Lie algebra su(n) of *SU*(*n*), so that (*T*0,*T*1,..., ) is a basis of *i* times the Lie algebra u(n) of the unitary group *U*(*n*) on C *n* . In those terms, we define *Tn* 2−1

$$T\_i^{(A)} = \frac{1}{|A|} \sum\_{\mathbf{x} \in \Lambda} T\_i(\mathbf{x}), \tag{10.272}$$

Finally, *h*˜ is a polynomial (which is sensitive to operator ordering). For example, to cast (10.270) (with *J* = 1) in the form (10.271), take *n* = 2, *Ti* = 1 2σ*i* (= 1,2,3), and

$$
\tilde{h}^{\rm CW}(T\_1, T\_2, T\_3) = -\mathcal{D}(T\_3^2 + BT\_1). \tag{10.273}
$$

The assumptions of Theorem 9.15 do not hold now, and indeed the local dynamics (9.40) fails to converge to global dynamics on the quasi-local C\*-algebra *A* defined by (8.130). Fortunately, it does converge to a global dynamics on the C\* algebra *C*(*S*(*B*)), where *B* = *Mn*(C) is the single-site algebra. In order to describe the limiting dynamics of (homogeneous) mean-field models as Λ % Z *d* , we equip the state space *S*(*B*) with the Poisson structure (8.52), which we now elucidate.

For unital C\*-algebras *B*, we may regard *S*(*B*) as a *w* ∗ -compact subspace of either the complex vector space *B* ∗ or the real vector space *B* ∗ sa; in the latter case we regard states as linear maps ω : *B* ∗ sa → R that satisfy ω(1*B*) = 1 and ω(*a* 2 ) ≥ 0 for each *a* ∈ *B*sa. If *B* = *Mn*(C), which is all we need, we may furthermore identify *B* ∗ sa with *i*u(n) ∗ , and since the value of each state ω ∈ *S*(*Mn*(C)) is fixed on *T*0 = 1*B* ∈ *i*u(n), it follows that *S*(*Mn*(C)) is a compact convex subset of *i*su(n) ∗ . In that case, the Poisson bracket (8.52) on *S*(*Mn*(C)) is none other than the restriction of (minus) the canonical Lie-Poisson bracket on su(n) ∗ ∼= *i*su(n) ∗ to *S*(*Mn*(C)), cf. (3.98) - (3.99). For example, for *n* = 2 we have *S*(*M*2(C)) ∼= *B*3 ⊂ R3 by Proposition 2.9, i.e.,

$$\mathfrak{a}\_{(\mathbf{x},\mathbf{y},\mathbf{z})}(a) = \operatorname{Tr}\left(\mathfrak{\rho}(\mathbf{x},\mathbf{y},\mathbf{x})a\right) \text{ (}(\mathbf{x},\mathbf{y},\mathbf{z}) \in \mathcal{B}^3, a \in \mathcal{M}\_2(\mathbb{C})\text{)};\qquad(10.274)$$

$$\rho(\mathbf{x}, \mathbf{y}, \mathbf{z}) = \frac{1}{2} \begin{pmatrix} 1+z \ \mathbf{x} - i\mathbf{y} \\ x+i\mathbf{y} \ 1-z \end{pmatrix}. \tag{10.275}$$

We also have su(2) ∗ ∼= R3 upon the choice of the basis (*Ti* = 1 2σ*i*), *i* = 1,2,3, of *i*su(2), which means that θ(*x*,*y*,*z*) ∈ *i*su(2) ∗ maps (*T*1,*T*2,*T*3) to (*x*, *y*,*z*) (where this time (*x*, *y*,*z*) ∈ R3), cf. §5.8). If we now regard the matrices *Ti* as functions *T*ˆ *i* on *B*3 by *T*ˆ *i*(ω) = ω(*Ti*), we find that the corresponding functions on *B*3 are given by

$$
\hat{T}\_1(\mathbf{x}, \mathbf{y}, \mathbf{z}) = \frac{1}{2}\mathbf{x}, \ \hat{T}\_2(\mathbf{x}, \mathbf{y}, \mathbf{z}) = \frac{1}{2}\mathbf{y}, \ \hat{T}\_3(\mathbf{x}, \mathbf{y}, \mathbf{z}) = \frac{1}{2}\mathbf{z}.\tag{10.276}
$$

The corresponding Poisson brackets (8.52) are {*T*1,*T*2} = −2*T*3 etc., i.e., {*x*, *y*} = −2*z* etc.; this is −2 times the bracket defined in (3.43) or (3.97) - (3.98). This factor 2 could have been avoided by moving to the three-ball with radius *r* = 1/2 instead of *r* = 1, whose boundary is the coadjoint orbit O1/2 naturally associated to spin- 1 2 .

We now return to our continuous bundle of C\*-algebras *A*(*c*) of Theorem 8.4, of course in the slightly adapted form appropriate to quantum spin systems, see §8.6. In particular, we recall that *A*(*c*) 0 = *C*(*S*(*B*)) and *A*(*c*) 1/*N* = *B*(*H*Λ*N* ), cf. (8.157) - (8.158), and hence we see the limit *N* → ∞ as a specific way of taking the limit Λ \$ Z*d* along the hypercubes Λ*N*. Symmetric and quasi-symmetric sequences (*a*1/*N*)*N*∈N are defined as explained after (8.161). The following observation is fundamental.

Theorem 10.22. *Let B* = *Mn*(C)*. If* (*a*1/*N*)*N*∈N *and* (*b*1/*N*)*N*∈N *are symmetric sequences with limits a*0 *and b*0 *as defined by* (8.46)*, respectively (so that* (*a*1/*N*)*N*∈N˙ *and* (*b*1/*N*)*N*∈N˙ *are continuous sections of the continuous bundle A*(*c*) *), then the sequence*

$$\left( \{a\_0, b\_0\}, i[a\_1, b\_1], \dots, i[\Lambda\_N | [a\_{1/N}, b\_{1/N}], \dots \} \right) \tag{10.277}$$

*defines a continuous section of A*(*c*) *. In particular, for each* ω ∈ *S*(*B*) *we have*

$$i\lim\_{N\to\infty}\mathfrak{o}^{|\Lambda\_N|}(|\Lambda\_N|[a\_{1/N},b\_{1/N}])=\{a\_0,b\_0\}(\mathfrak{o}).\tag{10.278}$$

*Proof.* The proof is a straightforward combinatorial exercise, and we just mention the simplest case where *d* = 1 and *a*1/*N* = *S*1,*N*(*a*1) and *b*1/*N* = *S*1,*N*(*b*1), where *a*1 ∈ *B* and *b*1 ∈ *B*, cf. (8.39). Then *a*0 = *a*ˆ1, *b*0 = *b*ˆ1, and similarly to (8.45) we find

$$\mathbb{E}\left[\mathcal{S}\_{1,N}(a\_1), \mathcal{S}\_{1,N}(b\_1)\right] = \frac{1}{N} \mathcal{S}\_{1,N}([a\_1, b\_1]),\tag{10.279}$$

Using (8.52), we find that (10.277) is equal to (*i*[ *a*1,*b*1],...,*S*1,*N*([*a*1,*b*1]),...). Since ω*N*(*S*1,*N*([*a*1,*b*1])) = ω([*a*1,*b*1]), the left-hand side of (10.278) is therefore equal to *i*ω([*a*1,*b*1]), which by (8.52) equals the right-hand side. - In other words, although the sequence of commutators [*a*1/*N*,*b*1/*N*] converges to zero (which is why *A* (*c*) 0 has to be commutative!), the rescaled commutators*iN*[*a*1/*N*,*b*1/*N*] converge to the macroscopic observable {*a*0,*b*0} ∈ *C*(*S*(*B*)). This reconfirms the analogy between the limit *N* → ∞ and the limit *h*¯ → 0 of Chapter 7, see especially Definitions 7.1 and 8.2. With *B* = *Mn*(C), Theorem 10.22 implies the central result about the macroscopic (and hence classical!) dynamics of mean-field theories:

Corollary 10.23. *Let* (*h*1/*N*)*N*∈N˙ *be a continuous section of A*(*c*) *defined by a symmetric sequence, and let* (*a*1/*N*)*N*∈N˙ *be an arbitrary continuous section of A*(*c*) *(i.e. a quasi-symmetric sequence). Then, writing h*1/*N* = *h*Λ*N for clarity, the sequence*

$$\left(a\_0(t), e^{ih\_{\Lambda\_1}t}a\_1e^{-ih\_{\Lambda\_1}t}, \cdots e^{ih\_{\Lambda\_N}t}a\_{1/N}e^{-ih\_{\Lambda\_N}t}, \cdots \right),\tag{10.280}$$

*where a*0(*t*) *is the solution of the equations of motion on S*(*Mn*(C)) *with classical Hamiltonian h*0 *and Poisson bracket* (8.52)*, defines a continuous section of A*(*c*) *.*

In other words, the Heisenberg dynamics on *A*Λ*N* = *B*(*H*Λ*N* ) defined by the quantum Hamiltonians *h*Λ*N* converges to the classical dynamics on the Poisson manifold *S*(*Mn*(C)) that is generated by their classical limit, viz. the Hamiltonian *h*0.

For example, since the operators *T* (Λ) *i* form symmetric sequences, so do Hamiltonians of the type (10.271). The limit *h*0 ∈ *C*(*S*(*Mn*(C))) of the family (*h*Λ ) in (10.271) is simply obtained by replacing the operators *T* (Λ) *i* in the function *h*˜ by the functions *T*ˆ *i* on *S*(*Mn*(C)). Equivalently, one may replace the *T* (Λ) *i* by the canonical coordinates (θ*i*) of *i*su(n) ∗ dual to the basis (*T*1,...,*Tn* 2−1 ) of *i*su(n) ∗ , i.e., θ*i*(*Tj*) = δ*i j*, and restricting the ensuing function on *i*su(n) ∗ to *S*(*Mn*(C)) ⊂ *i*su(n) ∗ .

Using (10.276), for the Curie–Weiss model (10.270) with *J* = 1 this gives

$$h\_0^{\rm CW}(\mathbf{x}, \mathbf{y}, \mathbf{z}) = -\frac{1}{2}\mathbf{z}^2 - \mathbf{B}\mathbf{x}.\tag{10.281}$$

The ground states of this Hamiltonian are simply its minima, viz.

$$\mathbf{x}\_{\pm} = (\mathcal{B}, \mathbf{0}, \pm\sqrt{1 - \mathcal{B}^2}) \text{ ( $0 \le \mathcal{B} < 1$ );} \tag{10.282}$$

$$\mathbf{x} = (1,0,0) \ (\mathbf{B} \ge 1),\tag{10.283}$$

all of which lie on the boundary *S* 2 of *B* 3 . Note that the points x± coalesce as *B* → 1, where they form a saddle point. Modulo our use of radius *r* = 1 instead of *r* = 1/2, this result coincides with (10.81) for classical limit of the quantum Ising model.

We now turn to symmetry and its possible breakdown. Suppose there is some subgroup of *U*(*n*), typically the image of a unitary representation *g* 7→ *ug* of a compact group *G* on C *n* , under which *h*˜(*T*0,*T*1,...,*Tn* 2−1 ) in (10.271) satisfies

$$
\tilde{h}(T\_0, \mu\_g T\_1 \mu\_g^\*, \dots, \mu\_g T\_{n^2 - 1} \mu\_g^\*) = \tilde{h}(T\_0, T\_1, \dots, T\_{n^2 - 1}) \tag{10.284}
$$

For example, in the Curie–Weiss model one has *G* = Z2, whose nontrivial element is represented by σ1. For (10.271) itself this implies *u* (*N*)*hN*(*u* (*N*) ) ∗ = *hN*, cf. (10.69).

Hence also in homogeneous mean-field models we obtain the structure (10.57), (10.58), and (10.59) familiar from the case of short-range forces. For the limit theory this implies that the classical Hamiltonian *h*0 on *S*(*Mn*(C)) is invariant under the coadjoint action of *G* ⊂ *U*(*n*) on *i*su(n) ∗ , restricted to *S*(*Mn*(C)) ⊂ *i*su(n) ∗ : in the Curie–Weiss model this "classical shadow" of the Z2 symmetry of the quantum theory is simply the map (*x*, *y*,*z*) → (*x*,−*y*,−*z*) on *B*3.

In the regime 0 < *B* < 1, the degenerate ground states of this model break this symmetry. In contrast, it can be shown from the Perron–Frobenius Theorem (which applies since both σ3 and σ1 are real matrices) that for *B* > 0 each quantummechanical Hamiltonian (10.270) has a unique ground state ψ(0) *N* . Being unique, this vector must share the invariance of *hN* under the permutation group S*N*, so that

$$\Psi\_N^{(0)} = \sum\_{n\_+=0}^N c(n\_+/N)|n\_+, n\_-\rangle,\tag{10.285}$$

where |*n*+,*n*− is the totally symmetrized unit vector in ⊗*N*C2 with *n*+ spins up and *n*− = *N* − *n*+ spins down, and *c* : {0,1/*N*,2/*N*,...,(*N* − 1)/*N*,1} → [0,1] is Frobenius Theorem). The asymptotic behaviour of *c* as *N* → ∞ has been studied, some function such that ∑*n*+ *c*(*n*+/*N*)2 = 1 (we may assume *c* ≥ 0 by the Perron– and as expected, *c* to converges pointwise to *c*(0) = *c*(1) = -1/2 and *c*(*x*) = 0, and zero elsewhere (at *B* = 0 one of course has either *c*(0) = 1 or *c*(1) = 1 for all *N*).

Thus we encounter a familiar headache: the "higher-level" theory *C*(*S*(*Mn*(C))) at *N* = ∞ breaks the Z2 symmetry, whereas the "lower-level" quantum theories *B*(*H*Λ*N* ) (*N* < ∞) do not, although the former should be a limiting case of the latter. Indeed, the situation for the Curie–Weiss model in the regime 0 < *B* < 1 is exactly analogous to the double-well potential as well as to the quantum Ising model in the same regime: if the two degenerate ground states x± ∈ *B*3 of *h*CW 0 are reinterpreted as Dirac measures δ± on *B*3, which in turn are seen as (pure) states ω± on the classical algebra of observables *C*(*S*(*M*2(C))), then (10.74) holds, *mutatis mutandis*.

The resolution of this problem through the restoration of Butterfield's Principle should also be the same as for the previous two cases: there is a first excited state ψ(1) *N* such that as *N* → ∞, the energy difference with the ground state approaches zero and one has approximate symmetry breaking as in (10.75)). Alas, for the Curie– Weiss model so far only numerical evidence is available supporting this scenario.

Equilibrium states of homogeneous mean-field models at any inverse temperature 0 < β < ∞ exist, despite the fact that in such models time-evolution α*t* on the infinite system *A* (and hence the KMS condition characterizing equilibrium states) is ill-defined (unless one passes to certain representations of *A*, which would be question-begging). Instead, one invokes the quasi-local C\*-algebra *A*, cf. (8.130), and *in lieu* of KMS states looks for limit points ωˆ β ∈ *S*(*A*) of the local Gibbs states ωβ Λ*N* defined by (9.96) as *N* → ∞; see (10.44) and surrounding discussion. Proposition 10.8 does not apply now, but Theorem 8.9 does: since each local Hamiltonian *h*Λ*N* is permutation-invariant (because each *T*(Λ*N*) *i* is), so is each local Gibbs state ωβ Λ*N* , and accordingly, each *w*∗-limit point of this sequence must share this property. As in (8.174), from the quantum De Finetti Theorem 8.9 we therefore have:

$$
\hat{\mathfrak{w}}^{\mathcal{B}} = \int\_{S(M\_n(\mathbb{C}))} d\mu\_{\mathfrak{F}}(\theta) \left( \mathfrak{w}\_{\theta}^{\mathcal{B}} \right)^{\infty}, \tag{10.286}
$$

for some probability measure µβ on the single-spin state space *S*(*Mn*(C)). By Proposition 8.28, this measure may also be regarded as a limit of the local Gibbs states, but now regarded as a state on the limit algebra *A* (*c*) 0 =*C*(*S*(*Mn*(C)))rather than as a state on *A* (*q*) 0 = *A*. By the same token, each state ω β θ in the decomposition (10.286) is a pure state on *A* (*c*) 0 (though seen as a state on *Mn*(C) it will be mixed!). The states ω β θ are computed as follows. Given a classical Hamiltonian *h*0 computed from (10.271) as explained after Corollary 10.23, for each point θ = (θ0,...,θ*n* 2−1 ) ∈ *i*u(n) ∗ we define a new self-adjoint operator *h*ˆ θ ∈ *Mn*(C) by

$$\hat{h}\_{\theta} = h\_0(\theta) \cdot 1\_n + \sum\_{i=0}^{n^2 - 1} \frac{\partial h\_0}{\partial \theta\_i}(\theta) \cdot T\_i. \tag{10.287}$$

For example, in the Curie–Weiss model, from (10.273) we have

$$h\_0^{\rm CW}(\theta) = -\Im(\theta\_3^2 + B\theta\_1);\tag{10.288}$$

$$
\hat{h}\_{\theta}^{\rm CW} = h\_0^{\rm CW}(\theta) - 2\theta\_3 \sigma\_3 - B\sigma\_1. \tag{10.289}
$$

Eq. (10.287) has the following origin. Let ω be any state on *A* for which the strong limit *T* (ω) *i* of each operator πω(*T* (Λ*N*) *i* ) on *H*ω exists as *N* → ∞ (for example, as in the proof of Theorem 8.16 one may show that this is the case when ω is a permutationinvariant state of *A*). It easily follows that *T* (ω) *i* lies in the algebra at infinity for πω, and hence in the center of πω(*A*) 00, cf. §8.5. If, in addition, ω is primary, then

$$T\_l^{(\mathfrak{a})} = \mathfrak{e}\_l \cdot \mathbf{1}\_{H\_{\mathfrak{a}}};\tag{10.290}$$

$$\Theta\_{\bar{l}} = \lim\_{N \to \infty} \mathcal{o}(T\_{\bar{l}}^{(\Lambda\_N)}). \tag{10.291}$$

Under these assumptions, we compute the commutator

$$
\left[\pi\_{\mathfrak{o}}(h\_{\Lambda\_{N}}), \pi\_{\mathfrak{o}}(a)\right] = \sum\_{i} \frac{\partial h\_{0}}{\partial \theta\_{i}} \left(T\_{0}^{(\Lambda)}, \dots, T\_{n^{2}-1}^{(\Lambda)}\right) \cdot \sum\_{\mathbf{x} \in \Lambda \mathcal{N}} \left[\pi\_{\mathfrak{o}}(T\_{i}(\mathbf{x})), \pi\_{\mathfrak{o}}(a)\right] + O\left(\frac{1}{|\Lambda\_{N}|}\right),
$$

where *a* ∈ ∪Λ *A*Λ , and *O*(1/|Λ*N*|) denotes a finite sum of (multiple) commutators between some power of *T* (Λ) *i* and operators that are (norm-) bounded in *N*. For example, for the Curie–Weiss model the *O*(1/|Λ*N*|) term is a multiple of

$$\sum\_{\boldsymbol{\chi}\in\Lambda\_{N}}[[\pi\_{\boldsymbol{\mathfrak{o}}}(\sigma\_{\boldsymbol{\mathfrak{o}}}(\boldsymbol{\chi})),\pi\_{\boldsymbol{\mathfrak{o}}}(\boldsymbol{a})],\sigma\_{\boldsymbol{\mathfrak{o}}}^{(A\_{N})}].\tag{10.292}$$

Since *a* is local, all commutators ∑*x*∈Λ*N* [πω(*Ti*(*x*)),πω(*a*)] are in πω(*A*), so that further commutators a la (10.292) vanish as ` *N* → ∞. Also, in this limit the terms *T*(Λ) *i* in the argument of ∑*i* ∂*h*0/∂ θ*i* assume their *c*-number values θ*i*, so that

$$\lim\_{N \to \infty} [\pi\_{\mathfrak{o}}(h\_{\Lambda\_N}), \pi\_{\mathfrak{o}}(a)] = [h\_{\mathfrak{o}}, \pi\_{\mathfrak{o}}(a)],\tag{10.293}$$

where formally (i.e. on a suitable domain) we have an ω-dependent Hamiltonian

$$h\_{\mathfrak{w}} = \sum\_{\boldsymbol{\chi} \in \mathbb{Z}^d} \pi(\hat{h}\_{\boldsymbol{\theta}}(\boldsymbol{\chi})),\tag{10.294}$$

where the θ*i* depend on ω via (10.291). Also, for each *a* ∈ *A* one has strong limits

$$\lim\_{N \to \infty} \pi\_{\mathfrak{o}\mathfrak{o}} \left( e^{ih\_{\Lambda\_N^t}t} a e^{-ih\_{\Lambda\_N^t}t} \right) = e^{ih\_{\mathfrak{o}\mathfrak{l}}t} \pi(a) e^{-ih\_{\mathfrak{o}\mathfrak{l}}t}.\tag{10.295}$$

Hence in the limit *N* = ∞ (provided it makes sense, which it does under the stated assumptions), the original mean-field Hamiltonian (10.271) with its homogeneous long-range forces converges to a sum of single-body Hamiltonians, in which the original forces between the spins have been incorporated into the parameters θ*i*.

Returning to (10.286), for any β = *T* −1, we now determine ωβ θ from the *Ansatz*

$$\alpha\_{\theta}^{\beta}(a) = \frac{\operatorname{Tr}\left(e^{-\beta\hat{h}\_{\theta}}a\right)}{\operatorname{Tr}\left(e^{-\beta\hat{h}\_{\theta}}\right)},\tag{10.296}$$

where θ is found by by solving the *self-consistency equation*

$$
\boldsymbol{\alpha}\_{\boldsymbol{\theta}}^{\beta} = \boldsymbol{\theta}.\tag{10.297}
$$

As explained after Corollary 10.23, here ωβ θ : *Mn*(C)sa → R is defined by its values on *i*su(n) and hence should be seen as a map *i*su(n) → R, like θ ∈ su(n) ∗ , so that (10.297) consists of *n*2 − 1 equations ωβ θ (*Ti*) = θ*i* (*i* = 1,...,*n*2 − 1). Alternatively, one may extend θ from *i*su(n) to *i*u(n) by prescribing θ(1*n*) = 1, and subsequently extend it further to *Mn*(C) by complex linearity. Clearly, the constant *h*0(θ) in (10.287) drops out of (5.152) and may be ignored in solving (10.297).

For example, if we take (10.289) with *B* = 0, then (10.297) forces θ1 = θ2 = 0, whereas the magnetization 2θ3 ≡ *m* = ωβ θ (σ3) satisfies the famous *gap equation*

$$\tanh(\mathcal{B}m) = m.\tag{10.298}$$

For any β this has a solution *m* = 0, i.e., θ = 0 in *B*3, which corresponds to the tracial state ω(*a*) = 1 2Tr(*a*) normally associated with infinite temperature (i.e., β = 0). This state is evidently Z2-invariant. For *T* ≥ *Tc* = 1/4 (i.e. β ≤ 4) this is the only solution. For *T* < *Tc* (or β > 4), two additional solutions ±*m*β (with *m*β > 0) appear, which break the Z2 symmetry. For *B* > 0 computations become tedious, but for β → ∞, where ωβ θ converges to the ground state of *h*ˆθ , one recovers our earlier conclusions.

#### Proposition 10.24. *The self-consistency equation* (10.297) *has at least one solution.*

*Proof.* This follows from Brouwer's Fixed Point Theorem (stating that any continuous map *f* from a compact compact set *K* ⊂ R*k* to itself has a fixed point), applied to *K* = *S*(*Mn*(C)) and *f*(θ) = ωβ θ , where θ ∈ *S*(*Mn*(C)), as just explained. -

The key result on equilibrium states of homogeneous mean-field theories, then, is:

Theorem 10.25. *Let h*Λ *in* (10.271) *define a homogeneous mean-field theory with compact symmetry group G. The sequence* (ωβ Λ*N* ) *of local Gibbs states defined by* (9.96) *and* (10.271) *has a unique G-invariant limit point* ωˆ β *, whose decomposition into primary states is given by* (10.286)*. The G-invariant probability measure* μβ *is concentrated on some G-orbit in S*(*Mn*(C))*, and the states* ωβ θ *on Mn*(C) *are given by* (10.296)*, with Hamiltonians h*ˆθ *defined by* (10.287)*, where* θ *satisfies* (10.297)*.*

*Proof.* We just sketch the proof, which is based on the Quantum De Finetti Theorem 8.9. Each operator *T*(Λ*N*) *i* is permutation-invariant, which property is transferred first to each local Hamiltonian *h*Λ*N* , thence to each local Gibbs state ωβ Λ*N* defined by *h*Λ*N* , and finally to each limit point of this sequence. As already noted, Theorem 8.9 then gives the decomposition (10.286), which by Theorem 8.29 (whose assumption holds in mean-field models) also gives the primary decomposition of ωˆ β (i.e., each state (ωβ θ )∞ is primary on the quasi-local algebra *A*). By our earlier argument centered on (10.294) - (10.295), time-evolution is implemented in the GNS-representation induced by such a state. An important step in the proof—which we omit because it requires various reformulations of the KMS condition we have not discussed—is that (ωβ θ )∞ satisfies the KMS condition with respect to the dynamics (10.295). This, in turn, implies (10.296), which, by definition of θ through (10.290) - (10.291), gives the self-consistency condition (10.297). The proof is completed by a tricky argument (which again uses alternatives to the KMS condition) to the effect that if some ωβ θ breaks the *G*-symmetry, the probability measure μβ on the *G*-orbit in *S*(*Mn*(C)) through ωβ θ induced by the normalized Haar measure on *G*, defines the only possible limit point of the local Gibbs states, and hence must be unique. -

Thus SSB can be detected by solving (10.297) and checking if the ensuing state(s) ωβ θ on *Mn*(C) is (are) *G*-invariant. As we have seen, in the Curie–Weiss model this is the case for β ≤ 4, whereas for β > 4 the measure μβ in (10.286) is given by

$$
\mu\_{\beta} = \frac{1}{2} (\delta\_{(0,0,m\_{\beta}/2)} + \delta\_{(0,0,-m\_{\beta}/2)}),
\tag{10.299}
$$

where δθ (*f*) = *f*(θ). In such cases, since each local Gibbs state is invariant, one faces the (by now) familiar threat to Earman's Principle. In response, we expect Butterfield's Principle to be restored through the introduction of asymmetric fleatype perturbations to *h*Λ that are localized in spin configuration space, although at nonzero temperature all excited states (rather than just the first) will start to play a role, and the precise details of the "flea" scenario remain to be settled.

#### 10.9 The Goldstone Theorem

So far, we have only discussed the simplest of all symmetry groups, namely *G* = Z2, which is both finite and abelian. Although it will not change our picture of SSB, for the sake of completeness (and interest to foundations) we also present a brief introduction to continuous symmetries, culminating in the Goldstone Theorem and the Higgs mechanism (which at first sight contradict each other and hence require a very careful treatment). The former results when the broken symmetry group *G* is a Lie group, whereas the latter arises when it is an infinite-dimensional gauge group.

Let us start with the simple case *G* = *SO*(2), acting on R2 by rotation. This induces the obvious action on the classical phase space *T*∗R2, i.e.,

$$R(p,q) = (Rp, Rq),\tag{10.300}$$

cf. (3.94), as well as on the quantum Hilbert space *H* = *L*2(R2), that is,

$$
\mu\_R \Psi(\mathbf{x}) = \Psi(\mathbf{R}^{-1}\mathbf{x}).\tag{10.301}
$$

Let us see what changes with respect to the action of Z2 on R considered in §10.1. We now regard the double-well potential*V* in (10.11) as an *SO*(2)-invariant function on R2 through the reinterpretation of *x*2 as *x*2 1+*x*2 2. This is the *Mexican hat potential*. Thus the classical Hamiltonian *h*(*p*,*q*) = *p*2/2*m*+*V*(*q*), similarly with *p*2 = *p*2 1+ *p*2 2, is *SO*(2)-invariant, and the set of classical ground states

$$\mathcal{E}\_0 = \{(p, q) \in T^\*\mathbb{R}^2 \mid p = 0, q^2 = a^2\} \tag{10.302}$$

is the SO(2)-orbit through e.g. the point (*p*1 = *p*2 = 0,*q*1 = *a*,*q*2 = 0). Unlike the one-dimensional case, the set of ground states is now connected and forms a circle in phase space, on which the symmetry group *SO*(2) acts. The intuition behind the Goldstone Theorem is that a particle can freely move in this circle at no cost of energy. If we look at mass as inertia, such motion is "massless", as there is no obstruction. However, this intuition is only realized in quantum field theory. In quantum mechanics, the ground state of the Hamiltonian (10.6) (now acting on *L*2(R2)) remains unique, as in the one-dimensional case. In polar coordinates (*r*,φ) we have

$$h\_{\hbar} = -\frac{\hbar^2}{2m} \left( \frac{\partial^2}{\partial r^2} + \frac{1}{r} \frac{\partial}{\partial r} + \frac{1}{r^2} \frac{\partial^2}{\partial \phi^2} \right) + V(r), \tag{10.303}$$

with *V*(*r*) = 1 4λ(*r*2 −*a*2)2. With

$$L^2(\mathbb{R}^2) \cong L^2(\mathbb{R}^+) \otimes \ell^2(\mathbb{Z}) \tag{10.304}$$

under Fourier transformation in the angle variable, this becomes

$$h\_{\hbar}\Psi(r,n) = \left(-\frac{\hbar^2}{2m}\left(\frac{\partial^2}{\partial r^2} + \frac{1}{r}\frac{\partial}{\partial r} - \frac{n^2}{r^2}\right) + V(r)\right)\Psi(r,n). \tag{10.305}$$

Since *h*¯ 2*n*2/2*mr*2 is positive, the ground state ψ(0) *h*¯ has ψ(0) *h*¯ (*r*,*n*) = 0 for all *n* = 0, and hence it is *SO*(2)-invariant, since the *SO*(2)-action on *L*2(R2) becomes

$$
\mu\_{\theta} \Psi(r, n) = \exp(in\theta) \Psi(r, n), \tag{10.306}
$$

after a Fourier-transform. Indeed, from a group-theoretical point of view, the unitary isomorphism (10.304) is nothing but the decomposition

$$L^2(\mathbb{R}^2) \cong \bigoplus\_{n \in \mathbb{Z}} H\_n,\tag{10.307}$$

where *Hn* = *L*2(R+) for all *n*, but with φ*n* ∈ *Hn* transforming under *SO*(2) as

$$
\mu\_{\theta} \phi\_n(r) = \exp(in\theta) \phi\_n(r) \ (\theta \in [0, 2\pi]). \tag{10.308}
$$

The *SO*(2)-invariant subspace of *L*2(R2), then, is precisely the space *H*0 in which ψ(0) *h*¯ lies. This is analogous to the situation occurring in one dimension higher (i.e. R3) with e.g. the hydrogen atom: in that case, the symmetry group is *SO*(3), and *L*2(R3) decomposes accordingly as

$$L^2(\mathbb{R}^3) \cong \bigoplus\_{j \in \mathbb{N}} H\_j;\tag{10.309}$$

$$H\_f = L^2(\mathbb{R}^+) \otimes \mathbb{C}^{2j+1}.\tag{10.310}$$

The ground state for a spherically symmetric potential, then, lies in *H*0 and is *SO*(3) invariant. For our purposes the relevant comparison is with the one-dimensional case: the decomposition of *L*2(R) under the natural Z2-action *u*−1ψ(*x*) = ψ(−*x*) is

$$L^2(\mathbb{R}) = H\_0 \oplus H\_1 \tag{10.311}$$

$$H\_l = \{ \Psi \in L^2(\mathbb{R}) \mid \Psi(\mathbf{x}) = (-1)^l \Psi(-\mathbf{x}) \}, \ i = 0, 1. \tag{10.312}$$

This time, *H*+ is the Z2-invariant subspace containing the ground state ψ(0) *h*¯ . Being Z2-invariant, ψ(0) *h*¯ is has peaks above both classical minima ±*a*; in fact, ψ(0) *h*¯ is realvalued and strictly positive. The ground state of the corresponding two-dimensional system, seen as an element of *L*2(R2), is just this wave-function ψ(0) *h*¯ extended from R to R2 by rotational invariance. Hence the ground state remains real-valued and strictly positive, with peaks about the circle of classical minima in R2.

Let us recall the situation for *d* = 1 (cf. §10.1). The first excited state ψ(1) *h*¯ lies in *H*1; it is real-valued, like ψ(0) *h*¯ , but since it has to satisfy ψ(1) *h*¯ (−*x*) = −ψ*h*¯(*x*), it cannot be positive. Indeed, with a suitable choice of phase, ψ(1) *h*¯ has one positive peak above *a* and the same peak but now negative below −*a*. Then the wave-function

$$
\Psi\_{\hbar}^{\pm} = (\Psi\_{\hbar}^{(0)} \pm \Psi\_{\hbar}^{(1)}) \sqrt{2}, \tag{10.313}
$$

is peaked above ±*a* alone (i.e., the negative peak of ±ψ(1) *h*¯ below ∓*a* exactly cancels the corresponding peak of ψ(0) *h*¯ ). The classical limit of ψ(0) *h*¯ comes out as the mixed state 1 2 (ω+ 0 + ω− 0 ), where ω± 0 = (*p* = 0,±*a*), but each state ψ± *h*¯ has the pure state ω± 0 as its classical limit. The latter are ground states, and hence in particular they are time-independent, because the energy difference *E*(1) − *E*(0) between ψ(1) *h*¯ and ψ(0) *h*¯ vanishes (even exponentially fast) as *h*¯ → 0.

A similar but more complicated situation arises in *d* = 2. The role of the pair

$$\left(\boldsymbol{\Psi}\_{\hbar}^{(0)} \in H\_0, \boldsymbol{\Psi}\_{\hbar}^{(1)} \in H\_1\right)$$

is now played by an infinite tower of unit vectors

$$\left(\boldsymbol{\Psi}\_{\hbar}^{(n)} \in H\_n, n \in \mathbb{Z}\right),$$

where ψ(*n*) *h*¯ is the lowest energy eigenstate (for *hh*¯ in (10.305)) in *Hn* ⊂ *L*2(R2). The analogue of the states ψ± *h*¯ for *d* = 1 involves a limit which heuristically is like

$$\lim\_{N \to \infty} \boldsymbol{\Psi}\_{\hbar}^{(N, \boldsymbol{\theta})} = \frac{1}{\sqrt{2N + 1}} \sum\_{n = -N}^{N} \boldsymbol{\mu}\_{\boldsymbol{\theta}} \boldsymbol{\Psi}\_{\hbar}^{(n)},\tag{10.314}$$

but this limit does not exist in *L*2(R2). As in §10.1, we instead rely on the technique explained around (10.4), which makes the unit vectors ψ(*N*,θ) *h*¯ converge to some probability measure μθ *h*¯ on R2 as *N* → ∞. In the subsequent limit *h*¯ → 0, one obtains a probability measure μθ 0 concentrated on a suitable point in the orbit of classical ground states (10.302). Similarly, in the same sense the ground state ψ(0) *h*¯ converges to a probability measure supported by all of E0.

To the extent that there is a Goldstone Theorem in classical mechanics, it would state that motion in the orbit E0 is free. That is, at fixed (*r* = *a*, *pr* = 0), where *pr* is the radial component of momentum, one has an effective Hamiltonian

$$h\_a(p\_\phi, \phi) = \frac{p\_\phi^2}{2ma^2},\tag{10.315}$$

whose time-independent states (*p*φ = 0,φ0) for arbitrary φ0 ∈ [0,2π) yield the ground states of the system, and whose "excited states"

$$(p\_\phi(t), \phi(t)) = \left(p\_\phi(0), \phi(0)\right) + \frac{p\_\phi(0)t}{ma^2}\right) \tag{10.316}$$

give motion along the orbit E0 with effective mass *ma*2, whose energy converges to zero as *p*φ → 0. However, since massless particles (whose existence is the main conclusion of the usual Goldstone Theorem) are not defined in classical mechanics, we now turn to relativistic field theory (with which we assume some familiarity).

We now illustrate SSB in classical field theory through a simple example, where the symmetry group is *G* = *SO*(*N*), but whenever write things down in such a way that the generalization to arbitrary scalar field theories is obvious. Suppose we have *N* real scalar fields ϕ ≡ (ϕ1,...,ϕ*N*), on which *SO*(*N*) acts in the defining representation on R*N*. Following the physics literature, from now on we sum over repeated indices like *i* and μ (*Einstein summation convention*). Let the Lagrangian

$$
\mathcal{L}^{\varrho} = \frac{1}{2} \partial\_{\mu} \mathfrak{q}\_{i} \partial^{\mu} \mathfrak{q}\_{i} - V(\mathfrak{q}), \tag{10.317}
$$

contain an *SO*(*N*)-invariant potential *V*, typically of the form (with ϕ2 ≡ ∑*N i*=1 ϕ2 *i* )

$$V(\boldsymbol{\varphi}) = -\frac{m^2}{2}\boldsymbol{\varphi}^2 + \frac{\lambda}{4}\boldsymbol{\varphi}^4,\tag{10.318}$$

where λ > 0, but *m*2 may have either sign. If *m*2 < 0, the minimum of *V* lies at ϕ = 0, but if *m*2 > 0 the minima form the *SO*(*N*)-orbit through

$$\mathfrak{g}^c = (\mathfrak{v}, \mathbf{0}, \cdots, \mathbf{0});\tag{10.319}$$

$$\nu \equiv m/\sqrt{\mathcal{A}} = ||\Phi^c||.\tag{10.320}$$

The idea is that the physical fields are excitations of the "vacuum state" ϕ*c*, so that, instead of ϕ, as the appropriate "small oscillation" field one should use

$$\mathcal{X}(\mathbf{x}) = \boldsymbol{\mathfrak{q}}(\mathbf{x}) - \boldsymbol{\mathfrak{q}}^{c}. \tag{10.321}$$

Consequently, the potential is expanded in a Taylor series for small χ as

$$V(\boldsymbol{\varphi}) = V(\boldsymbol{\varphi}^c) + \frac{1}{2}V\_{ij}''\chi\_i\chi\_j + O(\chi^3);\tag{10.322}$$

$$V\_{ij}^{\prime\prime} \equiv \frac{\partial^2 V}{\partial \!\!\! \! \! \! \! \! \! / \partial \! \! \! \! / \partial \! \! \! /} (\Phi^c). \tag{10.323}$$

Note that the linear term vanishes because *V* (ϕ*c*) = 0. We now use the *SO*(*N*) invariance of *V*, i.e., *V*(*g*ϕ) = *V*(ϕ) for all *g* ∈ *SO*(*N*). For *Ta* ∈ g (i.e. the Lie algebra of *G*, realized by anti-symmetric traceless *N* ×*N* matrices) this yields

$$\frac{d}{dt}V(e^{lT\_a}\boldsymbol{\varphi})\_{t=0} = 0 \Leftrightarrow \frac{\partial V(\boldsymbol{\varphi})}{\partial \boldsymbol{\varphi}\_l}(T\_a)\_{lj}\boldsymbol{\varphi}\_j = 0. \tag{10.324}$$

Differentiation with respect to ϕ*k* and putting ϕ = ϕ*c* then gives

$$V\_{ik}^{\prime\prime}(T\_a)\_{ij} \mathfrak{g}\_j^c = \mathbf{0}.\tag{10.325}$$

In general, let *H* ⊂ *G* be the stabilizer of ϕ*c*, i.e., *g* ∈ *H* iff *g*ϕ*c* = ϕ*c*. In our example (10.318) - (10.319), we evidently have *H* = *SO*(*N* − 1). Then *Ta*ϕ*c* = 0 for all generators *Ta* of the Lie algebra h of *H*, so that there are

$$M \equiv \dim(G) - \dim(H) = \dim(G/H) = \dim(G \cdot \mathfrak{g}^c) \tag{10.326}$$

linearly independent null eigenvectors of *V* (seen as an *N* ×*N* matrix). This number equals the dimension of the submanifold of R*N* where *V* assumes its minimum. In our example we have *M* = *N* −1, since dim(*SO*(*N*)) = 1 2*N*(*N* −1). We now perform an affine field redefinition, based on an affine coordinate transformation in R*N* that diagonalizes the matrix *V*. The original (real) fields were ϕ = (ϕ1,...,ϕ*N*), and the new (real) fields are (χ1,θ2,··· ,θ*N*), with

$$\mathcal{X}\_{\mathrm{l}} = \mathfrak{q}\_{\mathrm{l}} - \mathrm{v},\tag{10.327}$$

as in (10.321), and the *Goldstone fields* are defined, also in general, by

$$
\Theta\_a = \frac{1}{\nu} \langle T\_a \Phi^c, \Phi \rangle = \frac{1}{\nu} (T\_a)\_{ij} \Phi^c\_j \Phi\_i. \tag{10.328}
$$

Here ·,· denotes the inner product in R*N*, and we have chosen a basis of g in which the elements (*T*1,...,*T*dim(*H*)) form a basis of h, completed by *M* further elements (*T*dim(*H*)+1,...*T*dim(*G*)+1), so as to have basis of g. The index *a* in (10.328), then, runs from dim(*H*) +1 to dim(*G*), so that there are *M* Goldstone fields, cf. (10.326). In our running example, this number was shown to be *M* = *N* − 1, and in view of (10.319), the field θ*a* = (*Ta*)*i*1ϕ*i* is a linear combination of the ϕ2 till ϕ*N*.

The simplest example is *N* = 2, with potential (10.318) and *m*2 > 0. With the single generator *T* = −*i*σ2, we obtain θ = ϕ2. Since *V* = diag(2*m*2,0), we see that the mass term −1 2*m*2ϕ2 1 in (10.318) (with ϕ2 = ϕ2 1 + ϕ2 2 ) changes from the "wrong" sign −*m*2 to the 'right' sign +2*m*2 in (10.322), whilst −1 2*m*2ϕ2 2 in (10.318) disappears, so that the field θ comes out to be massless. Indeed, this is the point of the introduction of the Goldstone fields: in view of (10.325) and (10.328), the Goldstone fields do not occur in the quadratic term in (10.322) and hence they are massless, in satisfying a field equation of the form ∂μ ∂ μ θ*a* = ···, where ··· does not contain any term linear in any field. This proves the *classical Goldstone Theorem*:

Theorem 10.26. *Suppose that a compact Lie group G* ⊂ *SO*(*N*) *acts on N real scalar fields* ϕ = (ϕ1,...,ϕ*N*)*, leaving the potential V in the Lagrangian* (10.317) *invariant. If G is spontaneously broken to an unbroken subgroup H* ⊂ *G (in the sense that the stability group of some point* ϕ*c in the G-orbit minimizing V is H), then there are at least* dim(*G*/*H*) *massless fields, i.e., there is a field transformation*

$$(\mathfrak{q}\_1, \dots, \mathfrak{q}\_N) \mapsto (\mathfrak{X}\_1, \dots, \mathfrak{X}\_{N-M}, \mathfrak{G}\_1, \dots, \mathfrak{G}\_M) \text{ ( $M = \dim(G) - \dim(H)$ )}, \quad (10.329)$$

*that is invertible in a neighborhood of* ϕ = ϕ*c, such that the potential V*(ϕ)*, reexpressed in the fields* χ *and* θ*, has no quadratic terms in* θ*.*

The local invertibility of the field redefinition around ϕ*c* = 0 is crucial; in our example, where χ ≡ χ1 = ϕ1 −*v* and θ*a* = *Ta i*1ϕ*i*, this may be checked explicitly.

An alternative proof of Theorem 10.26 uses nonlinear Goldstone fields, viz.

$$\boldsymbol{\mathfrak{p}}(\boldsymbol{x}) = e^{\frac{1}{\nu}\theta\_{a}(\boldsymbol{x})T\_{a}}(\boldsymbol{\mathfrak{p}}^{c} + \boldsymbol{\mathcal{X}}(\boldsymbol{x})),\tag{10.330}$$

where the sum over *a* (implicit in the Einstein summation convention) ranges from 1 to *M*, *v* = ϕ*c*, and the fields χ = (χ1,...,χ*N*−*M*) are chosen orthogonal (in R*N*) to each *Ta*ϕ*c*, *a* = 1,...,*M*, and hence to the θ*a*. Provided that the generators of *SO*(*N*) (and hence of *G* ⊂ *SO*(*N*)) have been chosen such that

$$
\langle T\_a \mathfrak{p}^c, T\_b \mathfrak{p}^c \rangle = \nu^2 \mathfrak{d}^{ab}, \tag{10.331}
$$

the fields θ*a* defined by (10.330) coincide with the fields in (10.328) up to quadratic terms in χ and θ; to see this, expand the exponential and also use the fact that both *Ta*ϕ*c*,ϕ*c* and *Ta*ϕ*c*,χ vanish. This transformation is only well defined if *v* = 0, i..e., if SSB from *G* to *H* occurs, and its existence implies the Goldstone Theorem 10.26, for by (10.330) and *G*-invariance, *V*(ϕ) is independent of θ.

The Goldstone Theorem can be derived in quantum field theory, but in the spirit of this chapter we will discuss it rigorously for quantum spin systems. Far from considering the most general case, we merely treat the simplest setting. We assume that *A* is a quasi-local C\*-algebra given by (8.130), with *H* = C*n*. Furthermore:

$$
\gamma\_{\mathfrak{g}} \circ \mathfrak{a}\_{(\mathfrak{x},t)} = \mathfrak{a}\_{(\mathfrak{x},t)} \circ \gamma\_{\mathfrak{g}} \ ((\mathfrak{x},t) \in \mathbb{Z}^d \times \mathbb{R}, \mathfrak{g} \in G). \tag{10.332}
$$

$$G\_a = \{ \exp(sT\_a), s \in \mathbb{R}, T\_a \in \mathfrak{g} \}. \tag{10.333}$$

5. There is an *n*-tuple ϕ = (ϕ1,...,ϕ*n*) of local operators ϕα ∈ *Mn*(C) that transforms under *G* by ϕ → *ug*ϕ*u*∗ *g* = γ*g*(ϕ), and defines an order parameter φ*a* by

$$\mathfrak{d}\_a = \mathfrak{d}\_a \mathfrak{g} \equiv \frac{d}{ds} \left( \chi\_{\exp(sT\_a)}(\mathfrak{g}) \right)\_{|s=0},\tag{10.334}$$

at least for SSB of *Ga* (as above) in that, cf. Definition 10.6,

$$
\alpha(\delta\_a \varphi) \neq 0.\tag{10.335}
$$

6. Writing *j* 0 *a* = *iu* (*Ta*) ∈ *Mn*(C), it follows that δ*a*ϕ = −*i*[ *j* 0 *a*,ϕ], and hence that

$$\delta\_a \boldsymbol{\upvarphi}(\mathbf{x}) = -i \lim\_{\Lambda \nearrow \mathbb{Z}^d} \sum\_{\mathbf{y} \in \Lambda} [j\_a^0(\mathbf{y}), \boldsymbol{\upvarphi}(\mathbf{x})] \ (\mathbf{x} \in \mathbb{Z}^d), \tag{10.336}$$

since by (8.132) (i.e., Einstein locality) only the term *y* = *x* will contribute. Physicists then wish to define a charge by *Qa* = ∑*y*∈Z*d j* 0 *a*(*y*) and write (10.336) as δ*a*ϕ(*x*) = −*i*[*Qa*,ϕ(*x*)], but *Qa* does not exist precisely in the case of SSB!

Eq. (10.336) motivates the crucial assumption for the Goldstone Theorem, viz.

$$\mathfrak{so}(\delta\_a \mathfrak{g}(\mathbf{x}, t)) = -i \lim\_{\Lambda \nearrow \mathbb{Z}^d} \sum\_{\mathbf{y} \in \Lambda} \mathfrak{so}([j\_a^0(\mathbf{y}), \mathfrak{g}(\mathbf{x}, t)]) \text{ ( $\mathbf{x} \in \mathbb{Z}^d$ , t \in \mathbb{R})}, \qquad (10.337)$$

which incorporates the condition that the sum over *y* converge absolutely. Although (10.337) at first sight *softens* (10.336) in turning an operator equation into a numerical one, in fact (10.337) decisively *sharpens* (10.336) by involving the time-dependence of ϕ, whose propagation speed should be sufficiently small for enabling the limit in (10.336) to catch up with the limit in (10.337). As such, eq. (10.337) is satisfied with short-range forces, but the Meissner effect in superconductivity and the closely related Higgs mechanism in gauge theories (both of which circumvents the Goldstone Theorem) are possible precisely because in those cases (10.337) fails (at least in physical gauges, see also §10.10).

7. Finally, we make two assumptions just for convenience, namely

$$
\mathfrak{q}\_{\mathfrak{a}}(\mathfrak{x})^\* = \mathfrak{q}\_{\mathfrak{a}}(\mathfrak{x});\tag{10.338}
$$

$$
\mathfrak{o}(\mathfrak{q}\_{\mathfrak{a}}(\mathfrak{x})) = 0.\tag{10.339}
$$

If these are not the case, one could simply take real and imaginary components of ϕα and/or redefine ϕα as ϕ˜α = ϕα −ω(ϕα)· 1*A*, so that ω(ϕ˜α(*x*)) = 0.

The Goldstone Theorem provides information about the joint-energy momentum spectrum of the theory at hand. To define this notion, we exploit the fact that from assumption no. (3) and Corollary 9.12 we obtain a unitary representation *u*ω of the (locally compact) abelian space-time translation group *A* = Z*d* ×R on the GNSrepresentation space *H*ω induced by ω. The SNAG-Theorem C.114 applied to *A*, with dual *A*ˆ = T*d* ×R (cf. Proposition C.108), then yields a projection-valued measure

$$e\_{\mathfrak{a}} : \mathcal{\mathcal{O}}(\mathbb{R} \times \mathbb{T}^d) \to \mathcal{\mathcal{P}}(H\_{\mathfrak{a}}),\tag{10.340}$$

as a map from the Borel sets in R×T*d* to the projection lattice in *B*(*H*ω), such that

$$1\_{H\_{\mathfrak{A}}} = \int\_{\mathbb{T}^d} \int\_0^\infty de(E, k);\tag{10.341}$$

$$\mu\_{\mathfrak{U}}(\mathbf{y},t) = \int\_{\mathbb{T}^d} \int\_0^\infty de(E,k) \, e^{i(Et-\mathbf{y}\cdot\mathbf{k})} \text{ ( $\mathbf{y} \in \mathbb{Z}^d$ , t \in \mathbb{R})}.\tag{10.342}$$

Here *k* = (*k*1,...,*ld*), *y*·*k* = ∑*d i*=1 *yiki*, and we have reduced the integration range over *E* (which *a priori* would be R) to R+. Indeed, by Stone's Theorem we have *u*ω(*t*) = exp(*ith*ω), where σ(*h*ω) ⊂ [0,∞) because ω is a ground state by assumption, and the support of *e* is evidently contained in Z*d* ×σ(*h*ω) (cf. Definition A.16).

Definition 10.27. *The* joint energy-momentum spectrum σ(*h*ω, *p*ω) *of a spacetime invariant state* ω *(i.e.,* ω ◦ α(*x*,*t*) = ω*,* (*x*,*t*) ∈ Z*d* × R*) is the support of the projection-valued measure e*ω *associated to the* GNS*-representation* πω*, i.e., the smallest closed set* σ(*h*ω, *p*ω) ⊂ T*d* ×R *such that e*((T*d* ×R)\σ(*h*ω, *p*ω)) = 0*.*

The notation σ(*h*ω, *p*ω) is purely symbolic here, since (as opposed to the continuum case) the group Z*d* of spatial translations is discrete and hence has no generators *p*ω.

Since *u*ω(*x*,*t*)Ωω = Ωω, the origin (0,0) certainly lies in σ(*h*ω, *p*ω), with

$$e\_{\mathfrak{a}}(0,0) = |\mathfrak{Q}\_{\mathfrak{a}}\rangle\langle\mathfrak{Q}\_{\mathfrak{a}}|,\tag{10.343}$$

which by Theorem 9.14 is the unique T*d* × R-invariant state in *H*ω. Denoting this contribution to *e*ω by *e* (0) ω , in many physical theories one has *e*ω = *e* (0) ω +*e* (1) ω +···, where *e* (1) ω is supported on the graph of some continuous function *k* → ε*k* ≥ 0, i.e.,

$$\{(k, \mathfrak{e}\_k), k \in \mathbb{T}^d\} \subset \sigma(h\_{\mathfrak{a}\mathfrak{e}}, p\_{\mathfrak{a}\mathfrak{e}}) \subset \mathbb{T}^d \times \mathbb{R}.\tag{10.344}$$

The joint energy-momentum spectrum may be studied in part by considering

$$\begin{split} f(\mathfrak{e},p) &= \sum\_{\mathbf{y} \in \mathbb{Z}^d} \int\_{-\infty}^{\infty} dt \, e^{-i\varepsilon t + ip\cdot(\mathbf{x}-\mathbf{y})} \, \mathfrak{a}([J\_{\mathfrak{a}}^0(\mathbf{y}), \mathfrak{e}(\mathbf{x},t)]) \\ &= 2i \sum\_{\mathbf{y} \in \mathbb{Z}^d} \int\_{-\infty}^{\infty} dt \, e^{-i\varepsilon t + ip\cdot(\mathbf{x}-\mathbf{y})} \mathrm{Im} \langle \mathfrak{A}\_{\mathfrak{a}\mathfrak{a}}, \mathfrak{a}\_{\mathfrak{a}\mathfrak{a}}(j\_{\mathfrak{a}}^0(0)) e^{i\hbar h\mathfrak{a}} u\_{\mathfrak{a}\mathfrak{a}}(\mathbf{y}) \pi\_{\mathfrak{a}\mathfrak{a}}(\mathfrak{p}\_{\mathfrak{a}}(0)) \Omega\_{\mathfrak{a}\mathfrak{a}} \rangle \\ &= \int\_{\mathbb{T}^d} \int\_0^{\infty} \left( \langle \mathfrak{A}\_{\mathfrak{a}\mathfrak{a}}, \pi\_{\mathfrak{a}\mathfrak{a}}(j\_{\mathfrak{a}}^0(0)) de\_{\mathfrak{a}\mathfrak{a}}(E,k) \, \pi\_{\mathfrak{a}\mathfrak{a}}(\mathfrak{p}\_{\mathfrak{a}}(0)) \Omega\_{\mathfrak{a}\mathfrak{a}} \rangle \delta(\varepsilon - E) \delta(p - k) \\ & \quad - \langle \mathfrak{A}\_{\mathfrak{a}\mathfrak{a}} \pi\_{\mathfrak{a}}(\mathfrak{p}\_{\mathfrak{a}}(0)) de\_{\mathfrak{a}\mathfrak{a}}(E,k) \, \pi\_{\mathfrak{a}}(j\_{\mathfrak{a}}^0(0)) \Omega\_{\mathfrak{a}} \rangle \delta(\varepsilon + E) \delta(p + k) \right), \tag{10.345} \end{split}$$

i.e., the Fourier transform of the two-point function defined by *j* 0 *a* and ϕ, which is a distribution on the dual group T*d* × R; for the third equality we used a distributional version of the Fourier inversion formula (C.382). For example, if we replace *e*ω(*E*, *k*) by *e* (1) ω (*E*, *k*), then, since *e* (1) ω is absolutely continuous with respect to Haar measure *ddk* on T*d*, we see that *f*(ε, *p*) is proportional to δ(ε −ε*p*).

Theorem 10.28. *Under assumptions 1–7 (notably* (10.337) *and* SSB *of some continuous symmetry), the Hamiltonian h*ω *has continuous spectrum starting at zero and hence has no gap. If there is an excitation spectrum e*(1) ω *as explained above, with*

$$
\int \langle \Omega\_{\mathfrak{a}}, \pi\_{\mathfrak{a}}(j\_a^0(0)) de\_{\mathfrak{a}}^{(1)}(E, k) \, \pi\_{\mathfrak{a}}(\varphi\_{\mathfrak{a}}(0)) \Omega\_{\mathfrak{a}} \rangle \neq 0,\tag{10.346}
$$

*then the continuous function k* → ε*k defining the spectrum satisfies* ε0 = 0*.*

*Proof.* Since the sum in (10.337) converges absolutely, the Fourier transform ˇ*f*(*t*, *p*) of *y* → ω([ *j* 0 *a*(*y*),ϕ(*x*,*t*)]) in *y* alone is continuous in *p*, and by (10.337) we have

$$
\dot{a}a(\delta\_a \varphi(x,t)) = \check{f}(t,0). \tag{10.347}
$$

By (10.332), the left-hand side is independent of *x* and *t*, hence the Fourier transform *f*(ε,0) of the right-hand side in *t* is proportional to δ(ε). Since (10.343) does not contribute to *f* by (10.339), the calculation (10.345) shows that *f*(ε,0) = 0 if σ(*h*ω) has a gap. But *f*(ε,0) = 0 by (10.335), and so σ(*h*ω) has no gap. Similarly, for the final claim note that *f*(ε,0) ∼ δ(ε −ε0) as well as *f*(ε,0) ∼ δ(ε). -

#### 10.10 The Higgs mechanism

We proceed to a discussion of SSB in gauge theories, especially with an eye on the Higgs Mechanism, which plays a central role in the Standard Model of high-energy physics (whose empirical confirmation was more or less finished with the discovery of the Higgs boson at CERN, announced on July 4, 2012).

We look at the *Abelian Higgs Model*, given by the Lagrangian

$$\mathcal{L}^{\rho} = -\frac{1}{4} F\_A^2 + \frac{1}{2} \langle D\_\mu^A \Phi, D\_\mu^A \Phi \rangle - V(\Phi), \tag{10.348}$$

where ϕ = (ϕ1,ϕ2) is a scalar doublet, the usual electromagnetic field strength is

$$F\_{\mu\nu} = \partial\_{\mu}A\_{\nu} - \partial\_{\nu}A\_{\mu},\tag{10.349}$$

in terms of which *F*2 *A* = *F*μν*F*μν , and the covariant derivative is

$$D^A\_{\mu} \equiv \partial\_{\mu} - eA\_{\mu} \cdot T = \partial\_{\mu} \cdot 1\_2 + ieA\_{\mu} \cdot \sigma\_2. \tag{10.350}$$

Here *e* is some coupling constant, identified with the unit of electrical charge. We still assume that *V* only depends on ϕ2 = ϕ,ϕ and hence is *SO*(2)-invariant.

The novel situation compared to (10.317) and the like is that, whereas (10.317) is invariant under *global SO*(2) transformations, the Lagrangian (10.348) is invariant under *local SO*(2) *gauge transformations* that depend on *x*, namely

$$\mathfrak{g}(\mathbf{x}) \mapsto e^{\mathfrak{a}(\mathbf{x}) \cdot T} \mathfrak{g}(\mathbf{x}) = \begin{pmatrix} \cos \mathfrak{a}(\mathbf{x}) - \sin \mathfrak{a}(\mathbf{x}) \\ \sin \mathfrak{a}(\mathbf{x}) \ \cos \mathfrak{a}(\mathbf{x}) \end{pmatrix} \cdot \begin{pmatrix} \mathfrak{g}\_1(\mathbf{x}) \\ \mathfrak{g}\_2(\mathbf{x}) \end{pmatrix}; \quad (10.351)$$

$$A\_{\mu}(\mathbf{x}) \mapsto A\_{\mu}(\mathbf{x}) + \frac{1}{e} \partial\_{\mu} \mathfrak{a}(\mathbf{x}). \tag{10.352}$$

We say that the *local gauge group* G = *C*∞(R*d*,*U*(1)) acts on the space of fields (*A*,ϕ) by (10.351) - (10.352). Now suppose *V* has a minimum at some constant value ϕ*c* = 0. In that case, any field configuration

$$\boldsymbol{\varphi}(\mathbf{x}) = \exp(\boldsymbol{\alpha}(\mathbf{x}) \cdot \boldsymbol{T}) \boldsymbol{\varphi}^c;\tag{10.353}$$

$$A\_{\mu}(\mathbf{x}) = (1/e)\partial\_{\mu}\mathfrak{a}(\mathbf{x}) \mid (\mathfrak{a} \in \mathcal{G}),\tag{10.354}$$

minimizes the action. Hence the possible "vacua" of the model comprise the (infinite-dimensional) orbit V of the gauge group through (*A* = 0,ϕ = ϕ*c*). Note that *DA* μϕ = 0 for (*A*,ϕ) ∈ V , i.e., ϕ is *covariantly* constant along the vacuum orbit (whereas for global symmetries it is constant full stop). Relative to the (arbitrary) choice (0,ϕ*c*) ∈ V , we then introduce real fields χ and θ, called the *Higgs field* and the *would-be Goldstone boson*, respectively, by (10.330), which now simply reads

$$
\begin{pmatrix} \mathfrak{g}\_1(\mathbf{x}) \\ \mathfrak{g}\_2(\mathbf{x}) \end{pmatrix} = e^{\frac{1}{\nu}\theta(\mathbf{x}) \cdot T} \cdot \begin{pmatrix} \nu + \mathfrak{z}(\mathbf{x}) \\ \mathbf{0} \end{pmatrix} . \tag{10.355}
$$

After this redefinition of the scalar fields, the Lagrangian (10.348) becomes

$$\mathcal{A}\mathcal{E} = -\frac{1}{4}F\_B^2 + \frac{1}{2}\partial\_\mu \chi \partial^\mu \chi + \frac{1}{2}e^2(\nu + \chi)^2 B\_\mu B^\mu - V(\nu + \chi, 0), \tag{10.356}$$

where *B*μ = *A*μ −(1/*ev*)∂μ θ, and *F*2 *B* = *F*μν*F*μν for *F*μν = ∂μ*B*ν −∂ν*B*μ . This describes a vector boson *B* with mass term 1 2*m*2 *BB*μ*B*μ , with *m*2 *B* = 1 2 *e*2*v*2 > 0 (as opposed to the massless vector field *A*), and a scalar field χ with mass term 1 2*m*2 χ χ2, with *m*2 χ = (∂ 2*V*/∂ φ2 1 )|(*v*,0) > 0 (since *V* supposedly has a minimum at ϕ*c* = (*v*,0)).

This is the *Higgs mechanism*: the gauge field becomes massive, whilst the massless ("would-be") Goldstone boson disappears from the theory: it is (allegedly) "eaten" by the gauge field. Thus the scalar degree of freedom θ that seems lost is recovered as the longitudinal component of the massive vector field (which for a gauge field would have been an unphysical gauge degree of freedom, see below).

In the description just given, the Higgs mechanism in classical field theory is seen as a consequence of SSB. Remarkably, there is an alternative account of the Higgs mechanism, according to which it has nothing to do with SSB! Namely, we now perform a field redefinition analogous to (10.355) etc. straight away, viz.

$$
\begin{pmatrix} \mathfrak{q}\_1(\mathbf{x}) \\ \mathfrak{q}\_2(\mathbf{x}) \end{pmatrix} = e^{\mathfrak{q}(\mathbf{x}) \cdot T} \cdot \begin{pmatrix} \mathfrak{p}(\mathbf{x}) \\ \mathbf{0} \end{pmatrix}; \tag{10.357}
$$

$$A\_{\mu} = B\_{\mu} + (1/e)\partial\_{\mu}\theta. \tag{10.358}$$

This transformation is defined and invertible in a neighbourhood of any point (ρ0,θ0,*B*0, where ρ0 > 0, θ0 ∈ (−π,π), and *B*0 is arbitrary. Each of these new fields is gauge-invariant: for the gauge transformation (10.351) becomes

$$
\theta(\mathbf{x}) \mapsto \theta(\mathbf{x}) + \alpha(\mathbf{x});\tag{10.359}
$$

$$
\rho(\mathbf{x}) \mapsto \rho(\mathbf{x}),
\tag{10.360}
$$

and in view of (10.352), *B* does not transform at all. The Lagrangian becomes

$$\mathcal{L}^{\rho} = -\frac{1}{4} F\_B^2 + \frac{1}{2} \partial\_{\mu} \rho \, \partial^{\mu} \rho + \frac{1}{2} e^2 \rho^2 B\_{\mu} B^{\mu} - V(\rho), \tag{10.361}$$

with *V*(ρ) ≡ *V*(ρ,0). This is a Lagrangian without any internal symmetries at all (not even Z2, since ρ > 0), but of course one can still look for classical vacua that minimize the energy and hence the potential *V*(ρ). If ρ = 0 is the absolue minimum, then the above field redefinition is *a fortiori* invalidated, but if *V* (*v*) = 0 for some *v* > 0, we proceed as before, introducing a Higgs field χ(*x*) = ρ(*x*) − *v*, and recovering the Lagrangian (10.356). This once again leads to the Higgs mechanism.

This can be generalized to the nonabelian case; since it suffices to explain the idea, we just discuss the *SU*(2) case. In (10.348), the scalar field ϕ = (ϕ1,ϕ2) is now complex, forming an *SU*(2) doublet, the brackets ·,· now denote the inner product in C2, the nonabelian gauge field is *A* = *Aa*σ*a* (where the Pauli matrices σ*a*, *a* = 1,2,3, form a self-adjoint basis of the Lie algebra of *SU*(2)), with associated field strength *F*μν = ∂μ*A*ν −∂ν*A*μ +*g*[*A*μ ,*A*ν ] and covariant derivative *DA* μ = ∂μ +*igA*μ .

With *F*2 *A* = *Fa* μν*F*μν *a* , the Lagrangian (10.348) is invariant under the transformations

$$
\mathfrak{sp}(\mathfrak{x}) \mapsto e^{i\mathfrak{a}\_{\mathfrak{a}}(\mathfrak{x})\mathfrak{a}\_{\mathfrak{a}}(\mathfrak{x})} \mathfrak{q}(\mathfrak{x});\tag{10.362}
$$

$$A\_{\mu}(\mathbf{x}) \mapsto e^{i\mathbf{q}\_{\mathfrak{d}}(\mathbf{x})\mathfrak{G}\_{\mathfrak{d}}(\mathbf{x})}(A\_{\mu}(\mathbf{x}) - (i/\mathfrak{g})\partial\_{\mu})e^{-i\mathfrak{g}\_{\mathfrak{d}}(\mathbf{x})\mathfrak{G}\_{\mathfrak{d}}(\mathbf{x})}.\tag{10.363}$$

The definition of the gauge-invariant fields *B* and ρ a la (10.357) - (10.358) is now `

$$
\begin{pmatrix} \mathfrak{op}\_1(\boldsymbol{x}) \\ \mathfrak{p}\_2(\boldsymbol{x}) \end{pmatrix} = e^{i\boldsymbol{\theta}\_d(\boldsymbol{x}) \cdot \boldsymbol{\sigma}\_d} \cdot \begin{pmatrix} \mathfrak{p}(\boldsymbol{x}) \\ \mathbf{0} \end{pmatrix};\tag{10.364}
$$

$$A\_{\mu}(\mathbf{x}) = e^{i\theta\_{\mathrm{d}}(\mathbf{x})\sigma\_{\mathrm{d}}(\mathbf{x})} (B\_{\mu}(\mathbf{x}) - (i/\mathrm{g})\partial\_{\mu})e^{-i\theta\_{\mathrm{d}}(\mathbf{x})\sigma\_{\mathrm{d}}(\mathbf{x})},\qquad(10.365)$$

which leads, *mutatis mutandis*, to the very same Lagrangian (10.361).

As a compromise between these two derivations of the Higgs mechanism, it is also possible to fix the gauge by picking the representative (ϕ,*A*) in each G -orbit for which ϕ2(*x*) = 0 and ϕ1(*x*) > 0; note that this so-called *unitary gauge* is ill-defined if ϕ1(*x*) = 0. Calling this unique representative (ρ,*B*), we are again led to (10.361).

Gauge field theories are *constrained systems*, in which the *apparent* degrees of freedom in the Lagrangian are not the *physical* ones. For free electromagnetism, the Lagrangian is L (*A*) = −1 4*F*μν*F*μν , with *F*μν = ∂μ*A*ν − ∂ν*A*μ . In terms of the gauge-invariant fields *Ei* = *Fi*0 = ∂*iA*0 −∂0*Ai* and B = ∇×A, Maxwell's equations

$$\nabla \cdot \mathbf{E} = 0;\tag{10.366}$$

$$
\partial \mathbf{E}/\partial \mathbf{t} = \nabla \times \mathbf{B};\tag{10.367}
$$

$$\frac{\partial \mathbf{B}}{\partial t} = -\nabla \times \mathbf{E};\tag{10.368}$$

$$\nabla \cdot \mathbf{B} = 0,\tag{10.369}$$

then arise as follows: eqs. (10.366) and (10.367) correspond to the Euler–Lagrange equation for *A*0 and *Ai*, respectively, whereas (10.368) and (10.369) immediately follow from the definitions of B and E in terms of *A*. The Maxwell equations are in Hamiltonian form, with canonical momenta Πμ = ∂L /∂*A*˙μ ; this yields Π*i* = −*Ei*, as well as the *primary constraint* Π0 = 0. Nonetheless, the canonical Hamiltonian

$$h = \int d^3 \mathbf{x} \left( \Pi\_{\mu}(\mathbf{x}) \dot{\mathbf{A}}\_{\mu}(\mathbf{x}) - \mathcal{L}^{\rho}(\mathbf{x}) \right) = \int d^3 \mathbf{x} \left( \frac{1}{2} \mathbf{E}^2(\mathbf{x}) + \frac{1}{2} \mathbf{B}^2(\mathbf{x}) - A\_0(\mathbf{x}) \nabla \cdot \mathbf{E}(\mathbf{x}) \right),$$

is well defined. In the Hamiltonian formalism, Gauss' Law resurfaces as the *secondary constraint* stating that the primary constraint be preserved in time, viz.

$$\dot{H}\_0(\mathbf{x}) = -\frac{\delta h}{\delta A\_0(\mathbf{x})} = \nabla \cdot \mathbf{E}(\mathbf{x}) \equiv 0. \tag{10.370}$$

Since

$$\frac{d}{dt}\nabla \cdot \mathbf{E}(\mathbf{x}) = -\partial\_l(\delta h/\delta A\_l(\mathbf{x})) = -\partial\_l(\Delta A\_l - \partial\_l \nabla \cdot \mathbf{A}) = 0,\tag{10.371}$$

there are no "tertiary" constraints. Thus we have canonical phase space variables (E,A) and (Π0,*A*0), subject to (10.366) and to Π0(*x*) = 0 for each *x* ∈ R3, i.e.,

$$\Pi\_0(\lambda\_0) \equiv \int d^3x \,\Pi\_0(\mathbf{x})\lambda\_0(\mathbf{x}) = 0;\tag{10.372}$$

$$\Pi(\lambda) \equiv \int d^3x \nabla \cdot \mathbf{E}(\mathbf{x}) \lambda(\mathbf{x}) = 0,\tag{10.373}$$

for all (reasonable) functions λ0 and λ on R3. The constraints (10.372) - (10.373) are *first class* in the sense of Dirac, which means that their Poisson brackets are equal to existing constraints (or zero). In the Hamiltonian formalism, the role of the space-time dependent gauge transformations of the Lagrangian theory is played by the canonical transformations generated by the first class constraints, i.e.,

$$\delta\_{\lambda\_0} A\_0(\mathbf{x}) = \{\Pi\_0(\lambda\_0), A\_0(\mathbf{x})\} = \lambda\_0(\mathbf{x});\tag{10.374}$$

$$
\delta\_{\lambda\_0} A\_i(\mathbf{x}) = \delta\_{\lambda\_0} E\_i(\mathbf{x}) = 0;\tag{10.375}
$$

$$