Nonrelativistic Fields and Low-Energy Reduction

A nonrelativistic many-body theory is specified by more than a Hamiltonian density. One must identify the low-energy fields and their mass charges, the state or ensemble, the ultraviolet cutoff and matching data, the time-slicing and source conventions, and—if symmetry is broken—the density of commutators of broken charges. This chapter builds that specification from microscopic degrees of freedom and shows which conclusions survive changes of regulator or field variables. The state-dependent Goldstone rule used in the final layer is reviewed in Watanabe 2020, §§2–3.

Helpful background. Canonical Quantization: Algebra, Representation, and State supplies equal-time brackets, while Fock Space, Vacuum, and Particle Number supplies the occupation-number construction used in the first half of the chapter.

Enter this chapter

The common thread is the passage

$$\text{microscopic data} \longrightarrow \text{regulated fields} \longrightarrow \text{matched low-energy action} \longrightarrow \text{currents and collective modes}.$$

Each arrow has its own test. A continuum limit must retain the intended dispersion and state normalization; a coherent-state integral must retain its discrete-time prescription; a matched action must reproduce selected observables with errors suppressed by the breakdown scale; and a symmetry claim must satisfy the corresponding Ward identities. Generic Fock-space, symmetry, and effective-field-theory foundations remain with their earlier-volume treatments. Here the focus is their nonrelativistic realization.

This chapter primarily treats local, short-range systems with conserved particle number. Long-range forces, gauge fields, lattice point-group symmetries, explicitly open dynamics, and material-specific band structures require additional input. Unless stated otherwise, spatial dimension is $d$, $\hbar=1$, and a field of mass $m$ has free dispersion $\varepsilon_{\mathbf p}=\mathbf p^2/(2m)$.

Check your preparation

You can enter at different points. The following tests diagnose the shortest useful route.

Can you do this?	If yes	If not
Normalize creation operators in a box and recover a Dirac delta as $V\to\infty$	Begin with Galilean Fields, Scales, and Low-Energy Degrees of Freedom or go directly to the operator page	Repair with Fock Space, Vacuum, and Particle Number
Explain why a coupling can depend on a cutoff while a scattering amplitude cannot	Begin with continuum matching	Repair with Power Counting and Predictive Order
Derive a continuity equation from a localized phase rotation	Begin with currents and Ward identities	Repair with Localized Transformations and Ward–Takahashi Identities
Distinguish the number of broken generators from the number of gapless modes	Begin with finite-density Goldstone counting	Repair with Goldstone Counting, Low-Dimensional Obstructions, and Spacetime Exceptions

These are route choices, not a score. In particular, one can study operator normalization before learning the full coherent-state construction.

Choose a route

Goal	Suggested route	Result
First encounter	Galilean fields → second quantization → coherent-state integrals → microscopic reduction → power counting → currents	A regulated action with normalized observables and an error estimate
Symmetry and collective modes	Galilean fields → Schrödinger symmetry → currents → finite-density Goldstone counting	A distinction among exact, emergent, anomalous, and spontaneously broken symmetry
Many-body decouplings	Second quantization → coherent-state integrals → Hubbard–Stratonovich fields	An exact auxiliary-field identity separated from any saddle approximation
Effective-theory construction	Galilean fields → microscopic reduction → power counting	A matched operator hierarchy with a declared breakdown scale

The diagram makes the dependencies explicit. Solid arrows denote required construction steps; dashed arrows denote checks that can invalidate a proposed reduction.

A predictive nonrelativistic field theory requires normalized low-energy modes, a time-sliced functional representation, matched couplings, and source-defined observables before collective-mode claims are made. The diagram is schematic and not to scale; the dashed tests are logically independent.

Guide to the pages

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.

A chapter-wide convention and validity table

The table is the semantic comparison record for this chapter. A check in one row does not certify a different row.

Table: nonrelativistic constructions and their decisive tests.

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

Where reductions fail

The second figure organizes the most important negative tests. Follow a proposed field theory from its regulator choice to its final collective claim; a failure at any earlier node lowers the conclusion even if later algebra is internally consistent.

Time slicing, matching, cutoff stability, and source Ward identities test every construction shown. Scale-breaking or anomaly classification applies when scale symmetry is claimed, and the broken-charge rank applies when an internal symmetry is spontaneously broken. Passing one relevant test does not repair a failure of another. The map is schematic and not a numerical error estimate.

Review the chapter

Retrieval. State the boost phase of a field of mass $m$ and the commutator in which total mass appears as a central charge. A correct answer also states whether the transformation is active or passive.

Derivation. Starting from a nearest-neighbor lattice dispersion, recover the continuum effective mass and the first $O(a^2\mathbf p^4)$ correction. If the coefficient changes when the Fourier convention changes, the calculation has mixed a physical dispersion with notation.

Representation change. Write one normal-ordered quartic Hamiltonian as a discrete-time coherent-state integral, then integrate out a pairing auxiliary field to recover the quartic term. The endpoints, Grassmann sign, and Gaussian contour are part of the answer.

Failure diagnosis. A calculation finds two broken spin generators and announces two linear magnons. Inspect $\rho_{ab}$ before accepting it: nonzero magnetization pairs the generators and gives one type-B mode.

Synthesis. For a proposed dilute gas theory, give the fields, mass charges, state, cutoff, matched two-body observable, leading omitted operator, and source-defined number current. If scale symmetry or spontaneous internal-symmetry breaking is claimed, also give the scale-breaking classification or Goldstone data, respectively. Missing relevant information limits the claimed precision or symmetry content.

Continue

Quantum-Matter Correlators and Observable Conventions turns these source and field conventions into measurable spectral and response functions. Short-Range Scattering Data as Many-Body Inputs performs the first nonperturbative coupling match. Generic construction principles remain in Effective Field Theory as a Controlled Expansion.

References

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

Further reading

• Nishida, Yusuke, and Dam T. Son. “Nonrelativistic Conformal Field Theories.” Physical Review D 76 (2007): 086004. DOI. Open PDF.
• Son, Dam T., and Michael Wingate. “General Coordinate Invariance and Conformal Invariance in Nonrelativistic Physics: Unitary Fermi Gas.” Annals of Physics 321 (2006): 197–224. DOI. Open PDF.
• Watanabe, Haruki, and Hitoshi Murayama. “Unified Description of Nambu–Goldstone Bosons without Lorentz Invariance.” Physical Review Letters 108 (2012): 251602. DOI. Open PDF.