From Symmetry Breaking to Goldstone Theorem

Spontaneous symmetry breaking is the bridge from symmetry as classification to symmetry as low-energy dynamics.

In finite quantum mechanics, a symmetry often gives labels, degeneracies, and selection rules. In many-body physics and quantum field theory, a continuous global symmetry can be exact in the equations while the vacuum or phase fails to display it. Under suitable assumptions, that mismatch forces low-energy collective excitations: Goldstone modes.

The route connects a continuous global symmetry and its Noether current to a symmetry-breaking vacuum and, under the theorem’s hypotheses, a Goldstone mode.

This page explains the field-theory bridge. The conceptual prerequisites are Spontaneous Symmetry Breaking Preview and Goldstone Modes Preview. Goldstone Modes in Many-Body Systems develops nonrelativistic type-A/type-B counting and condensed-matter examples; the compact reference statement is Goldstone Theorem Preview.

From Symmetric Hamiltonian to Broken State

Let a continuous group $G$ act by unitary transformations on the theory. In ordinary Hamiltonian language, exact symmetry means

$$U(g)HU(g)^{-1}=H.$$

Spontaneous breaking means that the relevant state or vacuum is not invariant under the full group. Instead, it preserves only a subgroup $H\subset G$:

$$G \longrightarrow H$$

The broken directions are the elements of $G/H$. They label ways to move the order parameter without changing the energy at the perfectly symmetric level.

Here the subgroup $H$ is distinct from the Hamiltonian, despite the conventional repeated letter. Assume internal continuous symmetries and local dynamics in the discussion below. The finite-system caveat remains important. A finite system with a unique ground state usually preserves the exact symmetry as a ray. The broken-vacuum picture becomes sharp in an infinite-volume, thermodynamic, or field-theoretic limit.

Order Parameter and Broken Charge

An order parameter is an operator or field $\mathcal O$ whose expectation value diagnoses the symmetry breaking. A generator $Q_a$ is broken if the vacuum expectation value of the transformed operator is nonzero:

$$\langle\Omega|\delta_a\mathcal O|\Omega\rangle \ne 0.$$

Using the charge action

$$\delta_a\mathcal O = \frac{i}{\hbar}[Q_a,\mathcal O],$$

the same condition can be written as

$$\langle\Omega|[Q_a,\mathcal O]|\Omega\rangle \ne 0.$$

In a broken infinite-volume phase, $Q_a$ need not exist as an operator on that vacuum Hilbert space. Interpret the expression as the limit of a spatially smeared current charge acting in a commutator with a local observable. This limit can exist even though $Q_a|\Omega\rangle$ does not.

The Current Enters

For a continuous global symmetry in field theory, the charge is the spatial integral of a current density:

$$Q_a(t) = \int d^3x\,j_a^0(t,\mathbf x), \qquad \partial_\mu j_a^\mu=0.$$

Thus the broken-charge condition is really a statement about the integrated current:

$$\int d^3x\, \langle\Omega|[j_a^0(t,\mathbf x),\mathcal O(0)]|\Omega\rangle \ne 0.$$

A nonzero equal-time integral alone would not prove gaplessness. Current conservation makes the integrated commutator time independent when the surface term vanishes. Its spectral representation must then support arbitrarily low energies; in a Lorentz-invariant vacuum the associated pole is massless.

This is why Goldstone theorem is not just a picture of a tilted potential. It is a statement about current, charge, vacuum, and spectral weight.

Massless Pole Preview

In relativistic field theory, a broken continuous global symmetry is often visible as a pole in a current-order-parameter correlation function. Schematically,

$$\int d^4x\, e^{ip\cdot x} \langle\Omega| T\,j_a^\mu(x)\mathcal O(0) |\Omega\rangle \sim \frac{p^\mu F_a}{p^2+i0}$$

near $p^2=0$, when $\mathcal O$ overlaps with the broken direction. The pole represents a massless Goldstone boson.

The exact numerator, normalization, and operator choice are convention-dependent. The invariant lesson is that the current couples to a gapless excitation:

$$\langle\Omega|j_a^\mu(0)|\pi_b(p)\rangle \propto p^\mu.$$

The state $|\pi_b(p)\rangle$ is the field-theory particle or collective excitation associated with the broken generator. In condensed-matter language it may be a phonon, magnon, phase mode, or sound-like branch rather than a relativistic particle.

Mexican-Hat Picture and Its Limit

A common field-theory cartoon uses a complex scalar order parameter with potential

$$V(\phi) = \lambda \left( |\phi|^2-\frac{v^2}{2} \right)^2.$$

The minima satisfy

$$|\phi|=\frac{v}{\sqrt2}.$$

Choosing one phase breaks the global $U(1)$ symmetry. Writing fluctuations as

$$\phi(x) = \frac{1}{\sqrt2} \left[ v+h(x) \right] e^{i\pi(x)/v},$$

with $\lambda>0$ and $v>0$, the standard kinetic term $|\partial\phi|^2$ gives quadratic terms $\frac12(\partial h)^2+\frac12(\partial\pi)^2-\lambda v^2h^2$. Thus $m_h^2=2\lambda v^2$ and $m_\pi^2=0$ at tree level in $\hbar=c=1$ units.

This picture is useful, but it is not the theorem. The theorem does not require a literal two-dimensional potential drawn on a page. It requires the symmetry, broken vacuum, locality, and current assumptions.

Relativistic and Nonrelativistic Counting

For independent broken internal generators in a Lorentz-invariant vacuum under the standard Goldstone hypotheses, the number of Goldstone bosons equals the number of broken generators:

$$N_{\rm Goldstone} = N_{\rm broken}.$$

Nonrelativistic many-body systems can be subtler. Broken generators can form conjugate pairs in the ordered state, producing fewer Goldstone modes than broken generators. A common counting refinement uses

$$\rho_{ab} = -i\lim_{V\to\infty}\frac{1}{V} \langle[Q_a,Q_b]\rangle,$$

and, under suitable assumptions,

$$N_{\rm NG} = N_{\rm broken} - \frac12 \operatorname{rank}\rho.$$

This formula is included only as orientation. The detailed counting theorem belongs to many-body and field-theory treatments. For this bridge page, the key lesson is that continuous broken global symmetry produces gapless collective structure, while the exact number and dispersion can depend on the system.

Global Symmetry, Not Gauge Redundancy

Goldstone theorem applies to broken physical global symmetries. A gauge transformation is a redundancy in the description, not an operation that maps one physical state to a distinct physical state.

This distinction changes the spectrum. In a theory with a local gauge field, the would-be Goldstone field can combine with the gauge field. In particle-physics language this is the Higgs mechanism; in superconductors it is tied to electromagnetic response and the Anderson mechanism.

The safe bridge statement is:

$$\text{broken global symmetry} \quad \text{gives Goldstone modes,}$$

but

$$\text{gauge redundancy} \quad \text{requires a different analysis.}$$

The global-to-gauge distinction is developed in From Phase Symmetry to Gauge Theory.

Explicit and Approximate Breaking

If the symmetry is explicitly broken, the current is no longer exactly conserved:

$$\partial_\mu j^\mu \ne 0.$$

Then the Goldstone theorem’s exact gapless conclusion need not hold. If the explicit breaking is small, the mode may remain light but acquire a small gap. Such a mode is called a pseudo-Goldstone mode.

This is the field-theory version of a familiar idea from approximate symmetries: a small symmetry-breaking term turns an exact selection or degeneracy statement into an approximate one. The difference is that here the low-energy spectrum itself remembers the almost-flat broken direction.

Low-Dimensional and Long-Range Caveats

The assumptions matter. Continuous symmetry breaking can be obstructed by strong fluctuations in low dimensions. Coleman-type and Mermin–Wagner-type results show that certain low-dimensional relativistic or thermal systems cannot spontaneously break continuous symmetries under their stated hypotheses.

Long-range interactions, gauge fields, finite density, finite temperature, boundaries, and disorder can also change the simple conclusion or its observable form.

These caveats do not make the Goldstone theorem vague. They make it honest: the theorem is powerful because its assumptions are specific.

References

• A. Altland and B. Simons, Condensed Matter Field Theory, 2nd ed., Cambridge University Press, 2010.
• P. W. Anderson, “Plasmons, gauge invariance, and mass,” Physical Review 130, 439-442, 1963.
• S. Coleman, “There are no Goldstone bosons in two dimensions,” Communications in Mathematical Physics 31, 259-264, 1973.
• P. Coleman, Introduction to Many-Body Physics, Cambridge University Press, 2015.
• J. Goldstone, “Field theories with superconductor solutions,” Il Nuovo Cimento 19, 154-164, 1961.
• J. Goldstone, A. Salam, and S. Weinberg, “Broken symmetries,” Physical Review 127, 965-970, 1962.
• Y. Nambu, “Quasi-particles and gauge invariance in the theory of superconductivity,” Physical Review 117, 648-663, 1960.
• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995.
• H. Watanabe and H. Murayama, “Unified description of Nambu–Goldstone bosons without Lorentz invariance,” Physical Review Letters 108, 251602, 2012. Author preprint.
• S. Weinberg, The Quantum Theory of Fields, Volume II: Modern Applications, Cambridge University Press, 1996.

Exercises

1. Broken charge criterion.

Suppose $\delta\mathcal O=(i/\hbar)[Q,\mathcal O]$ and $\langle\Omega|\delta\mathcal O|\Omega\rangle\ne0$. What does this say about the action of $Q$ on the vacuum?

Solution

It says that the charge acts nontrivially in the vacuum sector. If the vacuum were invariant under $Q$ in the relevant sense, the expectation value of a pure symmetry variation would vanish. A nonzero value of

$$\langle\Omega|[Q,\mathcal O]|\Omega\rangle$$

means that the symmetry transformation changes an observable order parameter in the vacuum. That is the broken-generator signal.

2. Goldstone versus radial mode.

In the complex-scalar parametrization

$$\phi(x) = \frac{1}{\sqrt2} \left[ v+h(x) \right] e^{i\pi(x)/v},$$

which field is the Goldstone candidate and why?

Solution

The phase field $\pi(x)$ is the Goldstone candidate. It moves the order parameter along the circle of symmetry-related minima. A uniform change of that phase is a symmetry transformation, so the exact global-symmetry limit gives no potential energy cost for the uniform phase direction. The radial field $h(x)$ changes the magnitude and is typically massive.

3. Why does explicit breaking gap a Goldstone mode?

Solution

Explicit breaking tilts or lifts the formerly flat symmetry direction. In current language, the current is no longer exactly conserved. In effective-field language, a potential term appears for the would-be Goldstone coordinate. If that potential has nonzero curvature in the formerly flat direction, small oscillations acquire a gap. Weak breaking can then produce a pseudo-Goldstone mode. Breaking a symmetry does not guarantee that every formerly gapless direction is lifted; residual symmetries or special couplings can leave some modes gapless.

4. Why is a gauge theory not handled by naive Goldstone counting?

Solution

Goldstone theorem counts broken physical global symmetries. Gauge transformations are redundancies, so gauge-related configurations do not represent distinct physical states. When a charged condensate is coupled to a gauge field, the would-be Goldstone degree of freedom can combine with the gauge field and alter the spectrum. One must analyze gauge-invariant observables and constraints rather than naively counting broken gauge generators.