Fabri-Picasso Theorem

Prerequisites: Symmetry Breaking, Canonical Quantum Field Theory. Full path (23 pages)

• Intuitive
• Concrete
• Abstract
• Proof

Intuitive

For a symmetry with a conserved charge, we usually expect the charge to act on the vacuum like on any other state. If the vacuum carries no charge, applying the charge operator to it gives zero. If the symmetry is spontaneously broken, the naive picture is that the charge turns one vacuum into another, equally good vacuum, like turning all spins of a ferromagnet to a new direction.

Fabri and Picasso pointed out that in an infinite system this naive picture cannot be right. The charge is the integral of a charge density over all of space. In a broken vacuum each region of space contributes the same nonzero amount when we act with the charge density, because the vacuum looks the same everywhere. Adding up these contributions over infinite space gives an infinite result. The "rotated vacuum" would have infinite norm, so it is not a state of the theory.

The theorem therefore leaves two options: either the charge annihilates the vacuum and the symmetry is unbroken, or the rotated vacuum does not exist in the Hilbert space. In the second case the different vacua live in separate Hilbert spaces that no physical operation connects. This is the precise meaning of spontaneous symmetry breaking in quantum field theory, and it is why the vacua of a broken symmetry are not superposed the way degenerate states in ordinary quantum mechanics are.

Concrete

Fabri and Picasso showed in 1966 that if the vacuum state $|0\rangle$ is invariant under translations, it follows that the vacuum state is invariant under the internal symmetry or there is no state corresponding to $\hat Q |0\rangle$ in the Hilbert space of the theory. ($\hat Q$ is the Noether charge that corresponds to the internal symmetry.)

Fabri and Picasso (1966) showed that if the vacuum state $|0\rangle$ is translationally invariant, then the vacuum is either invariant under the internal symmetry, $Q |0\rangle = 0$ , or there is no state corresponding to $Q |0\rangle$ in the Hilbert space. The second case corresponds to SSB. The symmetry is hidden in that there is no unitary operator to map a physical state to its symmetric counterparts; instead, the symmetry is (roughly speaking) a map from one Hilbert space of states to an entirely distinct space. This is usually described as ‘vacuum degeneracy’, although each distinct Hilbert space has a unique vacuum state. […]

http://publish.uwo.ca/~csmeenk2/files/HiggsMechanism.pdf

Abstract

Let $j^\mu(x)$ be a conserved current with charge $Q = \int d^3x\, j^0(t,\vec x)$, and let the vacuum be invariant under the unitary translation operators, $U(a)|0\rangle = |0\rangle$, where $U(a) j^0(x) U(a)^{-1} = j^0(x+a)$. Suppose $Q|0\rangle$ is a normalisable state. Since $Q$ is conserved and commutes with the momentum operators, translation invariance gives $$\langle 0| Q\, Q |0\rangle = \int d^3x\, \langle 0| j^0(t,\vec x)\, Q |0\rangle = \int d^3x\, \langle 0| j^0(t,\vec 0)\, Q |0\rangle .$$ The integrand does not depend on $\vec x$, so the integral is either zero or infinite. A finite norm forces $\langle 0|QQ|0\rangle = 0$, hence $Q|0\rangle = 0$. This is the Fabri-Picasso theorem: either $Q|0\rangle = 0$, or $Q|0\rangle$ is not in the Hilbert space.

The precise formulation uses local charges. With smooth test functions, define $Q_R = \int d^4x\, f_R(\vec x)\, \alpha(t)\, j^0(x)$, where $f_R$ equals 1 for $|\vec x| < R$ and vanishes for $|\vec x| > 2R$, and $\alpha$ is a smooth function of time with integral 1. For a broken symmetry $\|Q_R|0\rangle\|$ grows without bound as $R\to\infty$. By locality, however, commutators $[Q_R, A]$ with any local operator $A$ become independent of $R$ for large $R$. The symmetry therefore always exists as a derivation $\delta(A) = \lim_{R\to\infty} [Q_R, A]$ of the algebra of local observables, and generates an automorphism of that algebra. What fails in the broken case is its unitary implementation in the vacuum representation.

In the language of algebraic quantum field theory, a spontaneously broken symmetry is an automorphism of the quasi-local algebra that maps the vacuum state to a different pure vacuum state. Distinct pure states that are invariant under translations and satisfy the cluster property give unitarily inequivalent representations by the GNS construction. Superselection between them is automatic: no local observable has nonzero matrix elements between the two vacuum sectors. The order parameter of the broken phase is $\langle 0|\delta(\phi)|0\rangle \neq 0$ for some local field $\phi$, and this is the input of the Goldstone theorem. In a finite volume the charge is a well-defined operator on the Hilbert space, so the second alternative of the theorem only arises for infinitely many degrees of freedom.

Proof

See the footnote 4 in http://publish.uwo.ca/~csmeenk2/files/HiggsMechanism.pdf

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1. Calculus
2. Imaginary Numbers
3. Linear Algebra
4. Functional
5. Group Theory
6. Hilbert Space
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8. Newton's Second Law
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19. Hamiltonian Formalism
20. Symmetry Breaking
21. Schrödinger Equation
22. Canonical Quantum Mechanics
23. Canonical Quantum Field Theory