Vacuum Expectation Value

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

Intuitive
Concrete
Abstract
Why is it interesting?

Intuitive

In quantum field theory the vacuum is the state of lowest energy. It is not empty in the everyday sense: quantum fields fluctuate in it, and some fields can have a nonzero average value everywhere in it. The vacuum expectation value of a quantity is its average value in this vacuum state, the value you would find by averaging many measurements made on the vacuum.

For most fields the average is zero. The electric field in empty space points in every direction with equal probability, so its average vanishes. The Higgs field is the important exception. Its lowest-energy configuration is not at zero but at a fixed nonzero value, like a ball resting in the circular trough at the bottom of a wine bottle, away from the bump in the middle. Since the vacuum fills all of space, this nonzero value is present everywhere. Particles that interact with the Higgs field feel this background, and this is how the W and Z bosons, quarks and charged leptons get their masses. A nonzero vacuum value of a field that transforms under a symmetry means that the vacuum itself does not respect that symmetry, which is what physicists call spontaneous symmetry breaking.

Vacuum expectation values have a second, broader use. Almost everything we compute in quantum field theory, from scattering probabilities to the propagation of particles, can be written as the vacuum average of a product of field operators. These averages, called correlation functions, are the basic output of the theory.

Concrete

The vacuum expectation value (VEV) of an operator $\mathcal{O}$ is

$\langle \mathcal{O}\rangle = \langle 0|\mathcal{O}|0\rangle ,$

where $|0\rangle$ is the normalized ground state of the full Hamiltonian. Two consequences of the symmetries of the vacuum restrict which VEVs can be nonzero. If $|0\rangle$ is invariant under translations, $U(a)|0\rangle = |0\rangle$ with $U(a)\phi(x)U(a)^{-1} = \phi(x+a)$ , then $\langle 0|\phi(x)|0\rangle = \langle 0|\phi(0)|0\rangle$ is a constant. If $|0\rangle$ is also Lorentz invariant, the VEV of a vector field must be an invariant vector, so $\langle 0|A_\mu(x)|0\rangle = 0$ . Only Lorentz scalars, such as a scalar field or a fermion bilinear $\bar\psi\psi$ , can have a nonzero one-point function.

The Higgs VEV. For the Higgs doublet with potential $V(\Phi) = -\mu^2\Phi^\dagger\Phi + \lambda(\Phi^\dagger\Phi)^2$ , $\mu^2, \lambda > 0$ , the classical minimum lies at $\Phi^\dagger\Phi = \mu^2/(2\lambda)$ . Using the gauge freedom we write the vacuum value as

$\langle\Phi\rangle = \frac{1}{\sqrt2}\begin{pmatrix} 0 \\ v \end{pmatrix}, \qquad v = \sqrt{\mu^2/\lambda} \approx 246\ \text{GeV}.$

Expanding around the VEV, $\Phi = \frac{1}{\sqrt2}(0, v + h)^T$ , the potential becomes $V = \text{const} + \lambda v^2 h^2 + \lambda v h^3 + \frac{\lambda}{4}h^4$ , so the Higgs boson has mass $m_h^2 = 2\lambda v^2 = 2\mu^2$ . The kinetic term $(D_\mu\Phi)^\dagger D^\mu\Phi$ evaluated on $\langle\Phi\rangle$ gives the gauge boson masses, for example $m_W = g v/2$ . In QCD the analogous order parameter is the quark condensate $\langle\bar q q\rangle \neq 0$ , a VEV of a composite operator, which breaks the approximate chiral symmetry.

Correlation functions. More generally the time-ordered VEVs

$G_n(x_1,\ldots,x_n) = \langle 0|T\,\phi(x_1)\cdots\phi(x_n)|0\rangle$

contain the physics of the theory. The LSZ reduction formula turns them into scattering amplitudes, and they are generated by the functional

$Z[J] = \langle 0|T\exp\left(i\int d^4x\, J(x)\phi(x)\right)|0\rangle, \qquad G_n = \frac{1}{i^n}\frac{\delta^n Z[J]}{\delta J(x_1)\cdots\delta J(x_n)}\Big|_{J=0},$

with $Z[0] = 1$ . When $\langle\phi\rangle = v \neq 0$ one usually shifts $\phi = v + \phi'$ and expands in the fluctuation $\phi'$ , whose one-point function vanishes.

No-Nonsense Quantum Field Theory by Jakob Schwichtenberg, section "Spontaneous Symmetry Breaking"

Abstract

In the algebraic formulation the vacuum is a state $\omega$ on the algebra of observables, a positive normalized linear functional, and the VEV of an observable $A$ is $\omega(A)$ . The GNS construction turns $\omega$ into a Hilbert space representation with cyclic vector $\Omega$ and $\omega(A) = \langle\Omega, \pi_\omega(A)\Omega\rangle$ . The Wightman functions $\langle\Omega, \phi(x_1)\cdots\phi(x_n)\Omega\rangle$ determine the theory completely by the reconstruction theorem.

Effective action. Define $W[J] = -i\ln Z[J]$ , the generating functional of connected correlation functions. The classical field $\phi_{\text{cl}}(x) = \delta W/\delta J(x)$ is the VEV of $\phi$ in the presence of the source. The Legendre transform

$\Gamma[\phi_{\text{cl}}] = W[J] - \int d^4x\, J(x)\phi_{\text{cl}}(x), \qquad \frac{\delta\Gamma}{\delta\phi_{\text{cl}}(x)} = -J(x),$

is the effective action, the generating functional of one-particle-irreducible vertices. At $J=0$ the VEV is a stationary point of $\Gamma$ . For constant fields $\Gamma[\phi_{\text{cl}}] = -\int d^4x\, V_{\text{eff}}(\phi_{\text{cl}})$ , so the true vacuum value minimizes the effective potential. Quantum corrections can shift the minimum or create one, as in the Coleman-Weinberg mechanism. Defined as a Legendre transform, $V_{\text{eff}}$ is convex; the non-convex potential found in perturbation theory describes the energy of homogeneous states, and in the region between degenerate minima the convex hull corresponds to mixtures of phases.

Spontaneous symmetry breaking. Let $Q$ generate a continuous internal symmetry, $\delta\phi = i[Q, \phi]$ . If $\langle\Omega, \delta\phi\,\Omega\rangle \neq 0$ for some field $\phi$ , the vacuum cannot be annihilated by $Q$ , and in infinite volume $Q\Omega$ is not even a normalizable vector: for the charge $Q_V$ of a finite volume $V$ , $\|Q_V\Omega\|^2 = \int_V d^3x\,\langle\Omega, j^0(x) Q_V\Omega\rangle$ , and by translation invariance this grows without bound as $V\to\infty$ . The symmetry is then spontaneously broken, the vacua related by the symmetry give unitarily inequivalent representations of the algebra of observables, and Goldstone's theorem implies massless excitations in a relativistic theory with local currents. In finite volume, tunnelling selects a unique symmetric ground state, and the nonzero VEV appears only after taking the infinite volume limit before switching off an explicit symmetry-breaking term.

Gauge theories. Elitzur's theorem states that a local gauge symmetry cannot be spontaneously broken: the VEV of a gauge non-invariant local operator such as $\Phi$ vanishes unless the gauge is fixed. The statement $\langle\Phi\rangle \neq 0$ in the Higgs mechanism therefore refers to a gauge-fixed formulation (for example unitary or $R_\xi$ gauges), where it is a convenient starting point for perturbation theory. Gauge-invariant quantities such as $\langle\Phi^\dagger\Phi\rangle$ are nonzero in any phase and are not order parameters by themselves.

Why is it interesting?

Vacuum expectation values are important to describe the Higgs mechanism and symmetry breaking in general.

In classical field theory, we are interested in the solutions of the Yang-Mills equations and these correspond to critical points of the action.

In contrast, in quantum field theory, we use the action to compute "vacuum expectation values" of observables. A vacuum expectation value is the average measured value in the vacuum state. The thing is that many physically interesting quantities can be expressed as vacuum expectation values through various tricks.

Vacuum Expectation Value

Intuitive

Concrete

Abstract

Why is it interesting?

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