spontaneously broken symmetry



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For ConfConf a configuration space and

S:Conf S : Conf \to \mathbb{R}

an action functional that is invariant under a group GG of symmetries acting on ConfConf, in that

gG,ϕConf:S(g(ϕ))=S(ϕ) \forall g \in G, \phi \in Conf : \,\,\, S(g(\phi)) = S(\phi)

a solution ϕ 0\phi_0 to the Euler-Lagrange equations of motion is said to exhibit spontaneously broken symmetry if it is not a fixed-point of that group action: if there is gGg \in G such that g(ϕ 0)ϕ 0g(\phi_0) \neq \phi_0.

The “breaking” refers to the fact that the group no longer acts. It is called “spontaneous” because one imagines that by a physical process the theory “finds” one of its solutions. This comes from the class of examples where a statistical system is first considered at high temperature and then cooled down. At some point it will “spontaneously” freeze in one allowed configuration. A standard example is a ferromagnet?: at high temperature its magnetization? vanishes, while at very low temperature it spontaneously finds a direction of magnetization, thus “breaking” rotational symmetry.

One calls the subgroup G ϕ 0GG_{\phi_0} \subset G that fixes the given configuration ϕ 0\phi_0 the unbroken symmetry group .

In the context of the quantum field theory arising by quantization of this action functional one considers the given classical solution ϕ 0\phi_0 as a background about which to consider perturbations of the remaining effective quantum field theory.

The fields in this effective QFT are then small excitations δϕ\delta \phi around the given ϕ 0\phi_0. Since the original symmetry group still acts on the full fields ϕ 0+δϕ\phi_0 + \delta \phi, the remaining symmetry group of the effective field theory is G ϕ 0G_{\phi_0}, whose elements gg send

g:(ϕ 0+δϕ)g(ϕ 0)+g(δϕ)=ϕ 0+gδϕ. g : (\phi_0 + \delta \phi) \mapsto g(\phi_0) + g(\delta \phi) = \phi_0 + g \delta \phi \,.

Since in the effective theory around ϕ 0\phi_0 the vacuum state where all the δϕ\delta \phi have no excitations (or rather: are in their ground state) corresponds to ϕ 0\phi_0 itself one says in this context that a quantum theory exhibits spontaneouly broken symmetry if its vacuum state is not invariant under the pertinent symmetries .

Formalization in cohesive homotopy-type theory

We indicate the formalization of the concept in the axiomatics of cohesion.

Let 𝔾\mathbb{G} be a cohesive abelian infinity-group (for instance 𝔾\mathbb{G} the circle group U(1)U(1) in smooth cohesion).

Then a prequantum line bundle on a phase space PP is given by a modulating morphism

PB𝔾 conn P \longrightarrow \mathbf{B}\mathbb{G}_{conn}

to the moduli stack B𝔾 conn\mathbf{B}\mathbb{G}_{conn} of 𝔾\mathbb{G}-principal connections.

A symmetry of the theory means that there is a cohesive infinity-group GG infinity-acting on PP such that the prequantum bundle descends to the homotopy quotient

P B𝔾 conn P/G \array{ P &\longrightarrow& \mathbf{B}\mathbb{G}_{conn} \\ \downarrow & \nearrow \\ P/G }

Now given an infinity-action of 𝔾\mathbb{G} on some VV (take 𝔾=U(1)\mathbb{G} = U(1) and V=V = \mathbb{C} for traditional quantum mechanics) then a quantum state (a wavefunction) is a section of the associated infinity-bundle, hence a diagram of the form

P Ψ V/𝔾 B𝔾 conn B𝔾 \array{ P &\stackrel{\Psi}{\longrightarrow}& V/\mathbb{G} \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ \mathbf{B}\mathbb{G}_{conn} &\longrightarrow& \mathbf{B}\mathbb{G} }

So this is something defined on phase space PP. If that also descends to the homotopy quotient P/GP/G (this is hard to draw here, but should be clear) then that makes the wavefunction also GG-equivariant. If not, then the wavefunction Ψ\Psi “breaks” the GG-symmetry.

Now if on top of this we have that the given Ψ\Psi is a “ground state”, then if it does not descend to the homotopy quotient we say “the GG-symmetry is spontaneously broken”.

To axiomatize what “ground state” means: introduce another \mathbb{R}-action on PP which is a Hamiltonian action, i.e. with respect to which the prequantum bundle is required to be equivariant. Then ask Ψ\Psi to (be polarized and) be a minimal eigenstate of the respective Hamiltonian. That makes it a “ground state”.

For more on the general translation between traditional geometric quantization and cohesive homotopy theory see at Higher geometric prequantum theory.


Scalars in mexican hat potential

A standard example which is both very simple but at the same time of central importance in one of the main applications in the standard model of particle physics – the electroweak symmetry breaking via the Higgs mechanism – is this:

Let Conf=C (X=[0,1] d, N)Conf = C^\infty(X = [0,1]^d, \mathbb{R}^N) be the configuration space of NN real scalar fields and take the action functional to be

S:ϕ X(12|ϕ| 2h2|ϕ| 2g4|ϕ| 4)dμ X S : \phi \mapsto \int_X \left( -\frac{1}{2}\vert \nabla \phi \vert^2 - \frac{h}{2} \vert\phi\vert^2 - \frac{g}{4} \vert\phi\vert^4 \right) d\mu_X

for some h,gh, g \in \mathbb{R}. This is manifestly invariant under the canonical action of the orthogonal group G=O(N)G = O(N) on ConfConf.

This action functional has a class of critical points given by constant maps ϕ:X n:ϕ(x)=Φ\phi : X \to \mathbb{R}^n : \phi(x) = \Phi. These extremize the action functional precisely if the Φ\Phi extremize the potential energy

h2|ϕ| 2+g4|ϕ| 4. \frac{h}{2} \vert\phi\vert^2 + \frac{g}{4} \vert\phi\vert^4 \,.

If both gg and hh are positive, then there is only one such critical point, given by Φ=0\Phi = 0. Therefore in this case the unique constant solution does not break the symmetry, in that g(Φ=0)=(Φ=0)g( \Phi = 0) = (\Phi = 0) for all gO(N)g \in O(N).

However, if hh is negative and gg positive, then the solutions are all those Φ\Phi with

|Φ| 2=hg>0. \vert \Phi \vert^2 = - \frac{h}{g} \gt 0 \,.

The set of all these is closed under the action of G=O(N)G = O(N) – this group takes one of these solutions into another – but none of these solutions is fixed by this action.

One says in this case that any such solution ϕ:xΦ\phi : x \mapsto \Phi is a solution that spontaneously breaks the symmetry of the theory.

In gravity

The theory of gravity on a given topological manifold XX has as configurations pseudo-Riemannian metrics on XX and its action functional – the Einstein-Hilbert action or one of its variants – is invariant under the action of the diffeomorphism group on XX.

The corresponding Euler-Lagrange equations are Einstein's equations. A given solution (X,g)(X,g) breaks the symmetry given by a diffeomorphism f:XXf : X \to X unless ff is an isometry. This means that the unbroken symmetries connected to the identity correspond precisely to the Killing vector fields on (X,g)(X,g).

Kaluza-Klein reductions

Spontaneous symmetry breaking in gravity plays a central role for instance in the context of the Kaluza-Klein mechanism. For instance for dimX=5dim X = 5 the effective field theory of gravity around a solution of the form (X=X 4×S 1,g 4g 1)(X = X_4 \times S^1, g_4 \otimes g_1) is 4-dimensional gravity coupled to electromagnetism (and a dilaton field): the components of the field of gravity along the circle transmute into the electromagnetic field. The ansatz breaks all the symmetries that would mix the remaining 4-dimensional gravitational excitations with these new electromagnetic excitations.

This is discussed in a bit of detail for instance in (Strominger, lecture 1).

Super Kaluza-Klein reductions

The above discussion has a direct analog in theories of higher supergravity. By the same logic, one finds that the effective quantum field theory around classical solutions that are Kaluza-Klein reductions of the form (X 4×Y d,(g 4g d))(X^4 \times Y^d, (g_4 \otimes g_d)) exhibits as global symmetries all those diffeomorphisms that are not spontaneously broken by this solution.

In this case, though, there are also supersymmetry analogs of the plain diffeomorphism action. Such a local supersymmetry remains unbroken in the given solution if it comes from a Killing spinor field.

Therefore KK-reductions to 4-dimensional Minkowski space in supergravity that admit precisely four Killing spinors of the form (ψ 4=constψ d=covariantlyconst)(\psi_4 = const \otimes \psi_d = covariantly const) give rise to effective field theories with exactly one remaining global supersymmetry. See also at supersymmetry breaking.

For more see supersymmetry and Calabi-Yau manifolds.

This is discussed in a bit of detail for instance in (Strominger, lecture 2).

Scherk-Schwarz mechanism

Specifically, the Scherk-Schwarz mechanism (Scherk-Schwarz 79) is the spontaneous supersymmetry breaking by KK-compactification on a circle whose spin structure imposes anti-periodic boundary conditions for fermion fields.



sponaneously broken global gauge group symmetry is discussed in vol I, section 19, and spontaneously broken local gauge group symmetry in vol I, section 21.4.


Textbook discussion of broken symmetry in gravity and supergravity in the context of the Kaluza-Klein mechanism is in

David Kazhdan, John Morgan, D.R. Morrison and Edward Witten, eds. , Quantum Fields and Strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999. (web version)

Discussion of spontaneous supersymmetry breaking is in

The article

points out that for symmetric systems with a symmetric ground state, already a tiny perturbation mixing the ground state with the first excited stated causes spontaneous symmetry breaking in the suitable limit, and suggests that this already resolves the measurement problem in quantum mechanics.