Nonrelativistic Fields and Low-Energy Reduction

Nonrelativistic field theory connects microscopic particles to low-energy collective behavior. The construction must preserve particle normalization, symmetries, and scattering observables before it can predict currents or gapless modes.

Enter this chapter

Begin with Galilean Fields, Scales, and Low-Energy Degrees of Freedom to identify the low-energy fields and their separation from microscopic scales.

Helpful background. Canonical Quantization: Algebra, Representation, and State and Fock Space, Vacuum, and Particle Number provide equal-time brackets and occupation-number states.

Topics

1. Galilean Fields, Scales, and Low-Energy Degrees of Freedom — Identifies mass as the central charge, derives the quadratic dispersion, and separates microscopic, cutoff, and many-body scales.
2. Schrödinger Symmetry, Scale Invariance, and Anomalies — Asks when $z=2$ scale invariance enlarges to the Schrödinger group and how renormalization introduces a scale.
3. Second-Quantized Bosons and Fermions — Fixes continuum and lattice normalization, statistics, number operators, and normal ordering.
4. Coherent-State Path Integrals for Many-Body Systems — Derives the discrete-time bosonic and Grassmann integrals before writing their continuum shorthand.
5. From Microscopic Hamiltonians to Continuum Fields — Gives a reproducible reduction near a band minimum and identifies the leading lattice corrections.
6. Hubbard–Stratonovich Fields and Collective Channels — Proves the Gaussian identity, compares density and pairing channels, and separates exact rewriting from saddle-point control.
7. Nonrelativistic Power Counting and Universality — Ranks operators about a named vacuum or finite-density scaling regime.
8. Densities, Currents, and Nonrelativistic Ward Identities — Defines observables by background-source variation and keeps contact terms and response limits visible.
9. Finite-Density Goldstone Counting — Derives the rank formula for broken internal charges and explains why a ferromagnet has one quadratic magnon for two broken generators.

How the ideas fit together

Second quantization and coherent-state integrals give complementary representations of the same regulated system. Matching then fixes the effective couplings from physical observables; introducing an auxiliary Hubbard–Stratonovich field is an exact rewriting, whereas retaining only its saddle is an additional approximation.

Finite density also changes the connection between symmetry and excitations. Broken internal charges can form conjugate pairs when their commutator has nonzero density, reducing the number of independent Goldstone modes Watanabe 2020, §§ 2–3. Counting broken generators alone therefore does not determine the low-energy spectrum.

The comparison below separates the assumptions and tests needed for these different conclusions.

A chapter-wide convention and validity table

Scroll horizontally to compare all columns. Use the Left and Right arrow keys when the table region is focused.

Construction, control, and failure tests in nonrelativistic field theory.

Construction or claim	Data that must be fixed	Invariant or observable check	Controlled domain	Characteristic failure
Galilean field	Species, mass charge, field normalization, dimension	Boosted dispersion changes by $\mathbf p\mapsto\mathbf p+m\mathbf v$	Momenta below the nonrelativistic breakdown scale	Mixing fields with different mass charges as one irreducible field
Second quantization	Box or continuum normalization; bosonic or fermionic brackets	Number eigenvalues and equal-time delta function	Chosen one-particle basis is complete in the retained sector	Losing a volume or lattice-spacing factor
Coherent-state integral	Normal ordering, time step, endpoints, thermal boundary condition	Exact single-mode partition function	Continuum notation taken only after the discrete kernel is fixed	Replacing the normal symbol by a naive classical Hamiltonian
Continuum reduction	Retained modes, matching observables, cutoff, symmetries	Agreement with microscopic dispersion and amplitudes through the claimed order	External scales small compared with the band or interaction cutoff	Double counting modes or silently restoring a broken symmetry
Hubbard–Stratonovich rewriting	Channel, sign, integration contour, normalization	Integrating out the auxiliary field reproduces the original interaction	Identity is exact; a saddle needs a separate small parameter	Treating a convenient channel or saddle as unique
Power-counted action	Fixed point, dynamical exponent, operator basis, breakdown scale	Cutoff stability after refitting the required couplings	Expansion parameters are demonstrably small or promoted terms are resummed	Using vacuum counting at a Fermi surface without translation
Current and response	Source sign, charge, contact terms, limit order	Continuity equation and Ward identity	Regulator and approximation preserve the stated symmetry	Omitting diamagnetic terms or interchanging $\omega\to0$ and $\mathbf q\to0$
Goldstone count	Broken charges, thermodynamic state, commutator densities	Rank of $\rho_{ab}=-i\langle[Q_a,Q_b]\rangle/V$ and low-momentum poles	Internal symmetries with the stated locality and limit hypotheses	Applying the internal-symmetry formula unchanged to spacetime or gauged symmetries

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Where to go next

Continue to Many-Body Correlators, Response, and Quasiparticles to match the measured observable to the appropriate correlator.

References

• Watanabe, Haruki. “Counting Rules of Nambu–Goldstone Modes.” Annual Review of Condensed Matter Physics 11 (2020): 169–187. DOI. Open PDF.