Schrödinger Symmetry, Scale Invariance, and Anomalies

A Galilean theory has Schrödinger symmetry only when its dynamics also admit dilatations with $z=2$ and special conformal transformations. Free fields and certain tuned fixed points do; a generic interaction, density, scattering length, or renormalization scale does not. In two-dimensional contact theories, renormalization produces dimensional transmutation, and the resulting contact term in the scale Ward identity is a quantum anomaly.

Required background. Galilean Fields, Scales, and Low-Energy Degrees of Freedom supplies boosts, mass charge, and $z=2$ kinematics.

Helpful background. The Conformal Algebra and Its Generators supplies the relativistic comparison, while What Is an Anomaly? distinguishes a quantum obstruction from explicit breaking.

The Schrödinger group

Work in flat space with a conserved positive mass operator $M$. The Schrödinger algebra adds a dilatation $D$ and special conformal generator $C$ to time translations $H$, spatial translations $P_i$, rotations $J_{ij}$, boosts $K_i$, and particle number. In a convention where

$$[D,H]=2iH, \qquad [D,C]=-2iC, \qquad [H,C]=-iD,$$

the triple $(H,D,C)$ forms $\mathfrak{sl}(2,\mathbb R)$. Boosts still satisfy $[K_i,P_j]=i\delta_{ij}M$. Changing generator signs changes all three displayed commutators together but not spectra or Ward identities.

For a primary operator $\mathcal O$ of mass $M_{\mathcal O}$ and scaling dimension $\Delta$, the transformations at the origin obey

$$[D,\mathcal O(0)]=i\Delta\mathcal O(0), \qquad [K_i,\mathcal O(0)]=[C,\mathcal O(0)]=0.$$

The nonrelativistic state–operator correspondence uses the oscillator Hamiltonian

$$H_\omega=H+\omega^2C.$$

A primary of dimension $\Delta$ creates a trapped state with energy $E=\omega\Delta$, subject to normalizability and a fixed mass sector. Nishida and Son prove this correspondence and use it for unitary few-body systems Nishida and Son 2007, §§ II–III. It is not the relativistic radial-quantization map: time scales twice as strongly as space, and mass superselection is essential.

Classical scaling and its hypotheses

The free action is invariant under

$$t\mapsto\lambda^2t, \qquad \mathbf x\mapsto\lambda\mathbf x, \qquad \psi(t,\mathbf x)\mapsto \lambda^{-d/2}\psi(\lambda^{-2}t,\lambda^{-1}\mathbf x).$$

A local operator $\mathcal O_i$ with scaling dimension $\Delta_i$ appears with coupling of dimension $d+2-\Delta_i$. Classical marginality is necessary but not sufficient for exact quantum scaling: the corresponding beta function must vanish, the measure must be invariant, and no boundary condition may introduce a scale.

For three-dimensional zero-range two-body scattering, the scattering length $a$ explicitly breaks scaling unless $a=0$ or $a^{-1}=0$. The unitary point $a^{-1}=0$ is Schrödinger invariant in the ideal zero-range limit; effective range, density, temperature, trap frequencies, and loss still introduce scales. Son and Wingate show how symmetry constrains the unitary-gas effective action without determining all coefficients Son and Wingate 2006, §§ 2–4.

Dimensional transmutation in two dimensions

Consider two distinguishable nonrelativistic fields in $d=2$ with a contact interaction. The bare coupling is classically dimensionless, but the two-body loop contains a logarithm. After renormalization, a convenient physical parametrization is a bound-state scale $E_B>0$, and the on-shell amplitude has the form

$$\mathcal A(E) =\frac{4\pi/m}{-\ln(E/E_B)+i\pi}$$

up to a convention-dependent overall sign. The pole at $E=-E_B$ and the energy dependence of the cross section are convention-independent checks. A dimensionless bare coupling has been replaced by a dimensionful observable: dimensional transmutation.

Let $g(\mu)$ multiply a renormalized contact operator $\mathcal O_C$. The local scale Ward identity has the structural form

$$2\,\mathcal H-\Pi^i{}_i =\beta_g(g)\,\frac{\partial\mathcal H}{\partial g} +\text{explicit breaking terms},$$

where $\Pi^i{}_i$ is the spatial stress trace in the declared source convention. The right-hand side is proportional to the contact density. Its coefficient depends on how $g$ and $\mathcal O_C$ are normalized; the measurable derivative of the energy with respect to $\ln E_B$ does not. Hofmann derives the anomalous commutator and contact contribution for the two-dimensional Fermi gas Hofmann 2012, Eqs. (10)–(16).

The anomaly is not “a regulator breaking the symmetry” if physical observables retain $E_B$ after the regulator is removed. A regulator that violates a symmetry spuriously is a separate problem. The test is whether counterterms can restore the Ward identity without erasing measured scale dependence.

Traps, breathing modes, and imperfect symmetry

The $\mathfrak{sl}(2,\mathbb R)$ algebra implies a ladder structure in an isotropic harmonic trap. In an exactly scale-invariant system the monopole or breathing mode has frequency $2\omega$. A shifted frequency or finite damping can arise from the anomaly, finite range, anisotropy, viscosity, or imperfect hydrodynamic conditions. Therefore a measured $2\omega$ oscillation is consistent with scale symmetry but does not alone prove the full operator algebra.

Pitaevskii and Rosch identified the hidden $SO(2,1)$ structure behind two-dimensional trapped breathing dynamics Pitaevskii and Rosch 1997, pp. R853–R856. In an interacting quantum gas, one must additionally check renormalization and the contact contribution.

Exact, emergent, anomalous, and explicitly broken

The distinctions are operational:

• exact symmetry: renormalized Ward identities hold with no breaking term;
• emergent symmetry: breaking operators are suppressed by powers of $Q/\Lambda_b$;
• anomalous symmetry: the classical transformation cannot be maintained together with renormalization, and a regulator-independent quantum term remains;
• explicit breaking: a parameter such as $a^{-1}$, $r_e$, $T$, $\mu$, or $\omega$ appears directly in the action or state.

Finite density is especially important: a Schrödinger-invariant Hamiltonian can be evaluated in a state with $\mu\ne0$. The theory’s operator symmetry and the state’s symmetry are different claims.

Common pitfalls

Equating $z=2$ with Schrödinger symmetry. Quadratic scaling does not guarantee boosts, special conformal transformations, or vanishing beta functions.

Calling every scale dependence anomalous. A scattering length or trap frequency is explicit breaking. An anomaly is the surviving quantum term after classical parameters and regulator artifacts have been accounted for.

Using the trap spectrum without its hypotheses. The state–operator map requires a primary in a fixed positive-mass sector and an exact Schrödinger-invariant fixed point. Generic finite-range Hamiltonians only approach it.

Exercises

Identify the scale-invariant scattering limits

In three dimensions the low-energy amplitude is $f(k)=1/(-a^{-1}-ik)$ at zero range. Show that continuous scaling can hold only at $a=0$ or $a^{-1}=0$.

Solution

Under $k\mapsto k/\lambda$, the term $ik$ scales as inverse length. A finite nonzero $a^{-1}$ supplies another inverse length and prevents homogeneous scaling. At $a^{-1}=0$, $f(k)=i/k$ is homogeneous; at $a=0$ the interaction disappears. These statements cease to be exact when effective range or another scale is retained.

Check the trapped ladder spacing

Using the displayed $\mathfrak{sl}(2,\mathbb R)$ commutators, construct operators proportional to $H-\omega^2C\pm i\omega D$ and show that their commutator with $H_\omega$ shifts energy by $\pm2\omega$.

Solution

Define $L_\pm=(H-\omega^2C\pm i\omega D)/(2\omega)$. Direct substitution gives $[H_\omega,L_\pm]=\pm2\omega L_\pm$ in the stated convention. Hence descendants occur in steps of $2\omega$; normalizability decides which ladder states exist.

Continue

Nonrelativistic Power Counting and Universality distinguishes fixed-point counting from finite-density counting. Short-Range Scattering Data as Many-Body Inputs performs the contact-coupling match explicitly. Universal Relations and Tan Contact turns the anomaly operator into measurable tails and thermodynamic derivatives.

References

• Hofmann, Johannes. “Quantum Anomaly, Universal Relations, and Breathing Mode of a Two-Dimensional Fermi Gas.” Physical Review Letters 108 (2012): 185303. DOI. Open PDF.
• Nishida, Yusuke, and Dam T. Son. “Nonrelativistic Conformal Field Theories.” Physical Review D 76 (2007): 086004. DOI. Open PDF.
• Pitaevskii, Lev P., and Achim Rosch. “Breathing Modes and Hidden Symmetry of Trapped Atoms in Two Dimensions.” Physical Review A 55 (1997): R853–R856. 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.