Quantization of Galilean Electrodynamics: a non-trivially trivial theory

Ander Urtiaga; Juan Miguel Nieto Garcia; Rafael Hernandez

REVIEW 4 major objections 5 minor 36 references

Quantization of Galilean Electrodynamics: a non-trivially trivial theory

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Galilean electrodynamics is fully constrained: it has zero propagating degrees of freedom, and its path integral contains only a single zero mode.

desk verdict A plausible zero-mode result for Galilean electrodynamics with a genuinely new constraint reclassification, but the path integral reduction that dismisses earlier propagators is not yet rigorous. read the letter →

arxiv 2608.01372 v1 pith:DOP5P42T submitted 2026-08-02 hep-th

Rafael Hernandez , Juan Miguel Nieto Garcia , Ander Urtiaga This is my paper · ORCID

classification hep-th

keywords GalileanelectrodynamicsconstrainedHamiltoniansystemsDirac-Bergmannalgorithmsecond-classconstraintspathintegralquantizationzeromodesnullreductionnon-relativisticgaugetheories

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes that Galilean electrodynamics — the non-relativistic limit of Maxwell theory obtained by null reduction — has zero propagating degrees of freedom. Running the Dirac-Bergmann constraint algorithm on the action, the authors find two first-class and six second-class constraints, which remove every dimension of the ten-dimensional phase space. The path integral built from the fully constrained system collapses to a single zero mode, and the only nonvanishing two-point function is $\langle \phi_e \phi_e \rangle \sim \delta^3(\mathbf{k})/\omega^2$, growing as $t^2$ in position space. If correct, the photon-like propagators reported in earlier quantizations are artifacts of applying Faddeev-Popov methods to a theory with second-class constraints and of only partially fixing the gauge. The result bears on the search for a non-relativistic holographic dual, since Galilean electrodynamics is the abelian sector of Galilean Yang-Mills.

What carries the argument

The load-bearing element is the Dirac-Bergmann constraint classification. Constraints are split into first-class (those that Poisson-commute with all other constraints) and second-class (those that obstruct naive quantization); the critical subtlety is identifying $\chi_5 = \partial_a \pi_a$ as first-class, which fixes the counting so that the phase space closes at zero dimensions. The second load-bearing element is the path-integral formula that imposes every constraint as a delta function — the extension of Faddeev-Popov to second-class systems — together with the boundary-condition input (compact support for the electric and magnetic fields, boundedness of $\pi_4$ at infinity) that turns the elliptic equations into delta functions on zero modes.

What would settle it

Find an admissible solution of the Galilean electrodynamics equations of motion — one with boundary conditions other than compact support of the electric and magnetic fields — whose inclusion in the path integral yields a nonvanishing $\langle A_a A_b \rangle$ or a second propagating mode; alternatively, repeat the Hamiltonian constraint analysis on a torus with periodic boundary conditions and obtain a nonzero number of physical degrees of freedom.

Watch

Extended reading notes

Core claim

The central claim is that Galilean electrodynamics is fully constrained: no field in the action is dynamical. In a ten-dimensional phase space, the Dirac-Bergmann algorithm yields two first-class constraints and six second-class constraints; the subtle step is recognizing that $\chi_5 = \partial_a \pi_a$ is first-class rather than second-class, after which the counting $10 - (2\cdot2) - (1\cdot6) = 0$ leaves no phase-space dimensions. With the gauge fixed by combining the Hamilton gauge with either the Lorenz or Coulomb gauge, the equations of motion become elliptic instead of hyperbolic, so plane waves are absent and only zero-momentum states exist. The path integral, written with delta functions for every constraint and integrated under the assumption that the electric and magnetic fields have compact support, reduces to $\delta(\partial_a J_a)\exp[(i/2)V\int dt\,dt'\, j_e(t)\,\partial_t^{-2} j_e(t')]$, producing $\langle \phi_e \phi_e \rangle \sim \delta^3(\mathbf{k})/\omega^2$ as the only nontrivial two-point function. The paper concludes that earlier nonvanishing propagators for this theory follow from quantizing with methods valid only for first-class constraints.

Load-bearing premise

The collapse of the path integral to a single zero mode assumes that the electric and magnetic fields have compact support and that $\pi_4$ is bounded at infinity; under other boundary conditions, additional harmonic solutions of the Laplace equations can enter and may change the propagator structure.

Editorial extensions

If this is right

If the counting is correct, the non-trivial propagators reported in earlier Faddeev-Popov treatments of Galilean electrodynamics are spurious; the theory's only physical correlator is the zero-mode $\langle \phi_e \phi_e \rangle \sim t^2$.
Galilean electrodynamics has no photon-like excitations: no plane waves, no momentum-carrying states, and the classical equations of motion are elliptic rather than hyperbolic.
Previous renormalization and interacting-field computations built on the old propagators would need to be redone with the zero-mode structure.
The path integral, and hence correlation functions, depends on boundary conditions at infinity; the theory is not topological even though it has no local propagating degrees of freedom.
The same analysis, applied to the quadratic sector of Galilean Yang-Mills, suggests that the non-abelian theory may also be fully constrained before interactions are turned on, although cubic and quartic terms could alter that.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A testable consequence of the zero-mode picture is that on a spatial torus or with periodic boundary conditions, additional harmonic modes may appear and act as discrete physical degrees of freedom; whether they propagate would probe the boundary-condition dependence of the claim.
If earlier propagator artifacts stem from second-class constraints, other null-reduced gauge theories (and Carrollian analogs) may hide similar zero-mode reductions behind apparently nontrivial Faddeev-Popov Green's functions; a constraint-classification pass before quantization would settle each case.
The $t^2$ growth of $\langle \phi_e \phi_e \rangle$ gives a sharp signature that could be looked for in lattice or Hamiltonian-truncation studies of Galilean electrodynamics, distinguishing the zero-mode theory from any theory with even a single propagating mode.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

You should know two things about this paper. First, the central constraint-counting result—that Galilean electrodynamics has zero propagating degrees of freedom—looks defensible, and the reclassification of χ5=∂aπa as a first-class constraint is a genuine step beyond earlier treatments. Second, the path integral that turns this into explicit zero-mode propagators is much less secure than the authors suggest, so the dismissal of [16,17] is not yet complete.

What is new: previous analyses (e.g. [12]) found one first-class and seven second-class constraints, which would give an odd-dimensional phase space. The authors fix this by noticing that χ5 is first-class, giving two first-class and six second-class constraints, and the dimension count 10−(2·2)−6=0 works. They then use the Senjanovic form of the path integral with both first- and second-class constraints. That is a real contribution, and the criticism that [16,17] apply Faddeev-Popov to a system with second-class constraints is legitimate.

The soft spots are in Section 4. The Dirac algorithm has a sign inconsistency: (3.11) gives {χ1,H1}=−χ3 while (3.13) gives +χ3. That is easy to fix, but it should be fixed. More serious is the transition from (4.3) to (4.4). The paper replaces functional deltas like δ(∂²ϕe) by deltas on finite-dimensional kernel coordinates q(t), p(t), g(x,t), and absorbs the Jacobian into an unspecified constant C. That is a formal manipulation, and it holds only under the compact-support and boundedness assumptions the authors acknowledge. The later g-integration is also not as clean as stated: g is restricted to harmonic functions, so ∫Dg exp(−i∫(∂aJa)g) imposes orthogonality to the harmonic subspace, not automatically the local condition ∂aJa=0. For compactly supported currents, completeness of harmonic polynomials probably saves the conclusion, but the paper does not say this. Because this step produces the zero-mode action (4.5) and the correlator δ³(k)/ω², the concrete propagator claims are conditional on a careful function-space argument.

The paper is honest about the boundary-condition dependence in Section 5, and the constraint count itself does not depend on those conditions. So the main structural claim is probably right; the path integral reduction needs more work.

Verdict: this deserves a serious referee. I would not cite it yet. For a reading group, maybe—it is a nice example of constrained quantization, but the unresolved functional-integral issues will distract.

Referee Report

4 major / 5 minor

Summary. This paper analyzes the Hamiltonian structure of Galilean electrodynamics (GED), the non-relativistic limit of Maxwell theory obtained by null reduction. The authors apply the Dirac-Bergmann algorithm, claim to find two first-class and six second-class constraints, and conclude that the phase space is reduced from ten to zero dimensions, i.e., the theory has no propagating degrees of freedom. They then solve the elliptic equations of motion under compact-support boundary conditions for the electric and magnetic fields, build a path integral with all constraints imposed as delta functions, and reduce it to a zero-mode action for a free quantum-mechanical particle. From this they derive the two-point function ⟨φeφe⟩ ∼ δ^3(k)/ω^2 and claim that it grows as t^2 in position space. The paper argues that earlier propagator computations in Refs. [16,17] are artifacts because those works apply Faddeev-Popov quantization, which the authors maintain is inapplicable to a system with second-class constraints.

Significance. If the zero-degree-of-freedom claim is correct, it is an important clarification of the physical content of GED: although the theory has non-trivial constraints and gauge symmetry, it contains no local propagating modes, and the previously reported non-trivial propagators would need reinterpretation. The paper's approach is parameter-free, and the constraint counting provides a concrete, checkable prediction. The authors also correctly emphasize the need to treat first- and second-class constraints simultaneously in the path integral, a point that is sometimes missed. The main limitation is that the path-integral reduction and the comparison with the literature contain technical gaps: the reduction of functional delta functions to zero modes, the treatment of the harmonic zero mode g, the evaluation of the Senjanovic determinants, and the claimed t^2 behavior all need further justification. The central counting result may survive these issues, but the quantum conclusions are not yet established.

major comments (4)

[Section 3.2, Eqs. (3.11), (3.13), (3.18)] Equation (3.11) defines χ3 through {χ1,H*1} = −∂a∂aφe = −χ3, but Eq. (3.13) states {χ1,H*2} = χ3. Since χ1 = π5 has vanishing Poisson bracket with each constraint added after H*1, the sign cannot flip; one of these equations is incorrect. The definition of the first-class combination χ5 = ∂aχ2,a + χ3 = ∂aπa used for the final counting is consistent only with one of the two signs, so the counting 10 − (2·2) − 6 = 0 needs to be re-derived with a correct and consistent constraint algebra. In addition, the transverse projector in Eq. (3.18) has the wrong sign: the standard projector that removes the longitudinal part is P^T_ab = δ_ab − ∂_a∂_b/∂_c∂_c. With the printed plus sign, the magnetic part in Eq. (3.20) is not reproduced, and the proposed solution u4,a = −P^T_ab A_b + \hat u_a with ∂c\hat u_c = π4 does not satisfy Eq. (3.15) unless ∂aAa = 0 is imposed prematurely. These are load-bearing issues because they underlie the first-class/second-class split and the final Hamiltonian Hf in Eq. (3.22).
[Section 4, Eqs. (4.3)–(4.5)] The reduction of the functional delta functions to finite-dimensional ones and the subsequent integration over g are not justified. After imposing ∫Dµ′ with δ(∂a∂aφe)δ(∂aAa)δ(3)(∂aπ4 − ∂bFba), the paper asserts, under compact support of E and B and boundedness of π4, that these become δ[φe−q(t)], δ[Aa−∂ag(x,t)], and δ(3)[π4−p(t)]. This is a change of variables in an infinite-dimensional integral; it requires a choice of function space, a nontrivial Jacobian, and a proof that the Laplacian kernels are exhausted by the stated zero-mode families. More importantly, the subsequent statement that 'the path integral is linear in g' and therefore integrating over g produces δ(∂aJa) is not correct as written: g is restricted to the harmonic subspace by ∂a∂ag = 0, so the integral of exp(i∫ g ∂aJa) over that subspace yields a delta functional on the annihilator of the harmonic functions, which is weaker than the local condition ∂aJa = 0. The paper supplies no argument that the harmonic test space together with the compact-support assumption forces the local delta. Because Eq. (4.5) is the entire basis for the correlator (4.6) and for the dismissal of Refs. [16,17], this step is load-bearing and must be made rigorous or replaced by an explicitly weaker statement.
[Section 4, Eqs. (4.1)–(4.4)] The Senjanovic-type measure in Eq. (4.1) contains the determinants |det({φa,ρb})| and |det({χa,χb})|^{1/2}. These determinants are absorbed into Dµ and never evaluated. If they are field-dependent, they contribute non-constant Jacobians under the change of variables (Aa,π4,φe) → (g,p,q) that leads to Eq. (4.4), and the constant C in (4.4) would not be a global factor. Since the final two-point function is extracted from the reduced path integral, the determinants need to be computed or shown to be constant; otherwise the normalization and the correlator (4.6) are not established. In particular, the bracket between the first-class constraint ∂aπa and the gauge condition ∂aAa is a field-dependent second-order differential operator.
[Section 4, Eq. (4.6) and following sentence] The position-space form of the two-point function is not t^2. With the standard distributional Fourier transform, ∫ dω/(2π) e^{−iωt}/ω^2 = −(1/2)|t| (up to the iε prescription and contact terms), so ⟨φeφe⟩ grows linearly in time, not quadratically. The claimed agreement with Ref. [15] for a field of conformal dimension Δ = 1 is therefore not obtained by a direct Fourier transform; a correlator ∼ t^2 would correspond to Δ = −1 in the usual normalization. This affects the interpretive claim of the paper and should be corrected.

minor comments (5)

[Section 3.4, Eq. (3.38)] The displayed Lorenz-gauge solution is incomplete because the condition ∂aAa + ∂tφe = 0 requires h′(t) = h(t); without this, the fields do not satisfy the gauge condition. State the relation explicitly.
[Section 4 and Section 3.2] There are minor typographical errors: 'trhe' after Eq. (4.2) should be 'the', and 'infered' in Section 3.2 should be 'inferred'.
[References] Reference [19] is listed as 'to appear' and should be updated or removed before publication.
[Eq. (3.18)] The inverse Laplacian 1/∂c∂c is used formally; specify the distributional or regularized definition, since the projector is later applied to field configurations.
[Section 5] The acknowledgement that the path integral depends on boundary conditions is appropriate, but the abstract and introduction state the zero-mode conclusion unconditionally; the authors should qualify the claim as 'no local propagating degrees of freedom under the stated boundary conditions' to avoid overstatement.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the zero-mode path integral is derived from the Dirac constraint analysis under explicitly stated boundary conditions, not from its own conclusion.

full rationale

The paper's central claim, that Galilean electrodynamics is fully constrained and has no propagating degrees of freedom, is obtained from a self-contained Dirac-Bergmann analysis in Section 3.2. The constraints (3.2), the secondary constraints (3.11)-(3.16), the recombination chi5 = ∂a chi2,a + chi3, and the phase-space counting 10 − (2·2) − (1·6) = 0 all follow from the Lagrangian (2.11) and its Poisson brackets; no external result or fitted parameter is used at this step. The path-integral reduction in Section 4 uses the Fradkin-Vilkovisky/Senjanovic formula (4.1) and then integrates out delta functions. The reduction to the zero-mode action (4.5) depends on the stated boundary conditions: compact support of the electric and magnetic fields and boundedness of π4 at infinity. These assumptions are explicit, and the paper acknowledges in Section 5 that different boundary conditions could change the result. This makes the conclusion conditional, but not circular: the paper does not define its conclusion into its premises. The comparison with the ⟨ϕeϕe⟩ two-point function of [15] is made after the derivation as a consistency check, not used as input. The only self-citations, notably the symmetry classification in [18] by Fontanella and Nieto García, are background material and are not load-bearing for the central claim. Concerns about whether functional deltas of differential operators can be replaced by finite-dimensional deltas, or about the integration over harmonic g producing δ(∂aJa), are mathematical correctness issues rather than instances of circular reasoning. No specific reduction of a prediction to a fitted input or to a self-citation chain can be exhibited, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The contribution is self-contained as an exercise in Dirac quantization once the standard GED action and the Senjanovic path integral are accepted. The only load-bearing extra input is the compact-support and boundedness boundary condition used to turn elliptic equations into zero modes. No free parameters or invented entities are introduced.

assumptions (4)

standard math The Dirac-Bergmann algorithm correctly classifies first- and second-class constraints.
Used throughout Section 3 as the standard constrained-system method; the paper relies on textbook statements from [26-28].
domain assumption Null reduction of five-dimensional Maxwell theory with Aμ independent of x5 yields the Galilean electrodynamics action (2.11).
Standard construction from [10,21]; the paper uses it to set up the theory, and the quantization claim is defined by the resulting action.
domain assumption The Senjanovic path integral formula (4.1) is valid for this mixed first- and second-class constrained system.
Quoted from [34,35]; its validity for field theories with these constraints is assumed without explicit proof.
ad hoc to paper Physical electric and magnetic fields have compact support and π4 is bounded at infinity.
Invoked in Section 4 to reduce Laplace-equation solutions to zero modes; the authors acknowledge in Section 5 that other boundary conditions could change the analysis.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Quantization of Galilean Electrodynamics: a non-trivially trivial theory." pith.science (2026). https://pith.science/paper/DOP5P42T

@misc{pith2026260801372,
  author       = {Pith},
  title        = {Pith review of: Quantization of Galilean Electrodynamics: a non-trivially trivial theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DOP5P42T}},
  note         = {Machine review of arXiv:2608.01372}
}

read the original abstract

We consider the quantization of the non-relativistic limit of electrodynamics called Galilean electrodynamics. To that end, we apply the Dirac bracket formalism to understand the constraints and dynamics of the theory and analyze its gauge structure. We show that the theory is fully constrained, which eliminates all the degrees of freedom from the path integral.

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

36 extracted references · 11 canonical work pages

[12]

Uniqueness of Galilean Conformal Electrodynamics and its Dynamical Structure

K. Banerjee, R. Basu and A. Mohan,Uniqueness of Galilean Conformal Electrodynam- ics and its Dynamical Structure,JHEP11(2019), 041 doi:10.1007/JHEP11(2019)041 [arXiv:1909.11993 [hep-th]]

work page Pith review arXiv 2019
[15]

Bagchi, R

A. Bagchi, R. Basu and A. Mehra,Galilean Conformal Electrodynamics, JHEP11 (2014), 061, doi:10.1007/JHEP11(2014)061, [arXiv:1408.0810 [hep-th]]

arXiv 2014
[1]

D. T. Son,Newton-Cartan geometry and the quantum Hall effect, Low Temp. Phys.51 (2025) no.3, 384-391, doi:10.1063/10.0035975, [arXiv:1306.0638 [cond-mat.mes-hall]]

arXiv 2025
[2]

Ciambelli, R

L. Ciambelli, R. G. Leigh, C. Marteau and P. M. Petropoulos,Carroll Structures, Null Geometry and Conformal Isometries, Phys. Rev. D100(2019) no.4, 046010, doi:10.1103/PhysRevD.100.046010 [arXiv:1905.02221 [hep-th]]

arXiv 2019
[3]

Fontanella,Non-extremal near-horizon geometries, Class

A. Fontanella,Non-extremal near-horizon geometries, Class. Quant. Grav.40(2023) no.13, 135006, doi:10.1088/1361-6382/acd980 [arXiv:2211.03861 [gr-qc]]

arXiv 2023
[4]

Lambert and J

N. Lambert and J. Smith,Non-relativistic M2-branes and the AdS/CFT correspon- dence, JHEP06(2024), 009, doi:10.1007/JHEP06(2024)009. [arXiv:2401.14955 [hep- th]]

arXiv 2024
[5]

Fontanella and J

A. Fontanella and J. M. Nieto Garc´ ıa,Constructing nonrelativistic AdS5/CFT4 holog- raphy, Phys. Rev. D111(2025) no.2, 026003, [erratum: Phys. Rev. D112(2025) no.2, 029902], doi:10.1103/PhysRevD.111.026003, [arXiv:2403.02379 [hep-th]]

arXiv 2025
[6]

Lambert and J

N. Lambert and J. Smith,Non-relativistic intersecting branes, Newton-Cartan geometry and AdS/CFT, JHEP07(2024), 224, doi:10.1007/JHEP07(2024)224, [arXiv:2405.06552 [hep-th]]

arXiv 2024

Show all 36 references

[7]

Fontanella and J

A. Fontanella and J. M. Nieto Garc´ ıa,Nonrelativistic Holography from AdS5/CFT4, Phys. Rev. Lett.133(2024) no.15, 151601, [erratum: Phys. Rev. Lett.135(2025) no.8, 089901] doi:10.1103/PhysRevLett.133.151601, [arXiv:2409.02267 [hep-th]]

2024 arXiv
[8]

Fontanella and O

A. Fontanella and O. Payne,A Carroll Limit of AdS/CFT: A Triality with Flat Space Holography?, Phys. Lett. B879(2026), 140669 doi:10.1016/j.physletb.2026.140669 [arXiv:2508.10085 [hep-th]]

2026
[9]

Fontanella and J

A. Fontanella and J. M. Nieto Garc´ ıa,An Introduction to String Newton-Cartan Holography and Integrability,[arXiv:2603.24657 [hep-th]]

arXiv
[10]

E. S. Santos, M. de Montigny, F. C. Khanna and A. E. Santana,Galilean covariant La- grangian models, J. Phys. A37(2004), 9771-9789, doi:10.1088/0305-4470/37/41/011

2004 doi
[11]

Le Bellac and J

M. Le Bellac and J. M. L´ evy-Leblond,Galilean electromagnetism, Nuovo Cim. B14 (1973) no.2, 217-234, doi:10.1007/BF02895715. 16

1973 doi
[13]

Festuccia, D

G. Festuccia, D. Hansen, J. Hartong and N. A. Obers,Symmetries and Couplings of Non-Relativistic Electrodynamics, JHEP11(2016), 037, doi:10.1007/JHEP11(2016)037, [arXiv:1607.01753 [hep-th]]

2016 arXiv
[14]

Bagchi, R

A. Bagchi, R. Basu, M. Islam, K. S. Kolekar and A. Mehra,Galilean gauge the- ories from null reductions, JHEP04(2022), 176, doi:10.1007/JHEP04(2022)176, [arXiv:2201.12629 [hep-th]]

2022 arXiv
[16]

Chapman, L

S. Chapman, L. Di Pietro, K. T. Grosvenor and Z. Yan,Renormalization of Galilean Electrodynamics, JHEP10(2020), 195, doi:10.1007/JHEP10(2020)195, [arXiv:2007.03033 [hep-th]]

2020 arXiv
[17]

Banerjee and A

K. Banerjee and A. Sharma,Quantization of interacting Galilean field theories, JHEP 08(2022), 066, doi:10.1007/JHEP08(2022)066, [arXiv:2205.01918 [hep-th]]

2022 arXiv
[18]

Fontanella and J

A. Fontanella and J. M. Nieto Garc´ ıa,Revisiting the symmetries of Galilean Electro- dynamics, JHEP06(2025), 058, doi:10.1007/JHEP06(2025)058, [arXiv:2411.19217 [hep-th]]

2025 arXiv
[19]

Fontanella and J

A. Fontanella and J. M. Nieto Garc´ ıa, to appear
[20]

Bergshoeff, J

E. Bergshoeff, J. Rosseel and T. Zojer,Non-relativistic fields from arbitrary con- tracting backgrounds, Class. Quant. Grav.33(2016) no.17, 175010, doi:10.1088/0264- 9381/33/17/175010, [arXiv:1512.06064 [hep-th]]

2016 arXiv
[21]

H. P. Kuenzle and C. Duval,Relativity and nonrelativistic physical theories on five- dimensional space-time., Class. Quantum Grav.3(1986), 957, doi:10.1088/0264- 9381/3/5/024

1986 doi
[22]

Susskind,Model of selfinduced strong interactions,Phys

L. Susskind,Model of selfinduced strong interactions,Phys. Rev.165(1968), 1535- 1546, doi:10.1103/PhysRev.165.1535

1968 doi
[23]

Takahashi,Towards the Many Body Theory With the Galilei Invariance as a Guide

Y. Takahashi,Towards the Many Body Theory With the Galilei Invariance as a Guide. 1, Fortsch. Phys.36(1988), 63-81

1988
[24]

Takahashi,Towards the Many Body Theory With the Galilei Invariance as a Guide

Y. Takahashi,Towards the Many Body Theory With the Galilei Invariance as a Guide. 2, Fortsch. Phys.36(1988), 83-96

1988
[25]

Omote, S

M. Omote, S. Kamefuchi, Y. Takahashi and Y. Ohnuki,Galilean Co- variance and the Schr¨ odinger Equation, Fortsch. Phys.37(1989), 933-950, doi:10.1002/prop.2190371203

1989 doi
[26]

P. A. M. Dirac, Lectures on Quantum Field Theory, 1964, Belfer Graduate School of Science, Yeshiva University. 17

1964
[27]

P. A. M. Dirac,Generalized Hamiltonian dynamics,Can. J. Math.2(1950), 129-148 doi:10.4153/CJM-1950-012-1

1950 doi
[28]

Thiemann,Modern Canonical Quantum General Relativity, Cambridge University Press, 2007

T. Thiemann,Modern Canonical Quantum General Relativity, Cambridge University Press, 2007

2007
[29]

Castellani,Symmetries in Constrained Hamiltonian Systems, Annals Phys.143 (1982), 357, doi:10.1016/0003-4916(82)90031-8

L. Castellani,Symmetries in Constrained Hamiltonian Systems, Annals Phys.143 (1982), 357, doi:10.1016/0003-4916(82)90031-8

1982 doi
[30]

J. M. Pons,On Dirac’s incomplete analysis of gauge transformations, Stud. Hist. Phil. Sci. B36(2005), 491-518, doi:10.1016/j.shpsb.2005.04.004

2005 doi
[31]

Palumbo,Quantization of Galilean gauge theoriesPhys

F. Palumbo,Quantization of Galilean gauge theoriesPhys. Rev. D30(1984), 2148 doi:10.1103/PhysRevD.30.2148

1984 doi
[32]

De Franceschi and F

G. De Franceschi and F. Palumbo,Dirac brackets for Galilean gauge theoriesPhys. Rev. D30(1984), 2156 doi:10.1103/PhysRevD.30.2156

1984 doi
[33]

Greiner and J

W. Greiner and J. ReinhardtField Quantization,Springer-Verlag, Berlin, 1996, ISBN 978-3-540-59179-5, 978-3-642-61485-9 doi:10.1007/978-3-642-61485-9

1996 doi
[34]

E. S. Fradkin and G. A. Vilkovisky,Quantization of relativistic systems with con- straints, Phys. Lett. B55(1975), 224-226, doi:10.1016/0370-2693(75)90448-7

1975 doi
[35]

P. Senjanovic,Path Integral Quantization of Field Theories with Second Class Con- straints, Annals Phys.100(1976), 227-261, [erratum: Annals Phys.209(1991), 248], doi:10.1016/0003-4916(76)90062-2

1976 doi
[36]

de Leeuw, A

M. de Leeuw, A. Fontanella and J. M. Nieto Garc´ ıa,A perturbative approach to the non-relativistic string spectrum,JHEP10(2024), 096 doi:10.1007/JHEP10(2024)096 [arXiv:2403.09563 [hep-th]]. 18

2024 arXiv

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [12]

Uniqueness of Galilean Conformal Electrodynamics and its Dynamical Structure

K. Banerjee, R. Basu and A. Mohan,Uniqueness of Galilean Conformal Electrodynam- ics and its Dynamical Structure,JHEP11(2019), 041 doi:10.1007/JHEP11(2019)041 [arXiv:1909.11993 [hep-th]]

work page Pith review arXiv 2019

[2] [15]

Bagchi, R

A. Bagchi, R. Basu and A. Mehra,Galilean Conformal Electrodynamics, JHEP11 (2014), 061, doi:10.1007/JHEP11(2014)061, [arXiv:1408.0810 [hep-th]]

arXiv 2014

[3] [1]

D. T. Son,Newton-Cartan geometry and the quantum Hall effect, Low Temp. Phys.51 (2025) no.3, 384-391, doi:10.1063/10.0035975, [arXiv:1306.0638 [cond-mat.mes-hall]]

arXiv 2025

[4] [2]

Ciambelli, R

L. Ciambelli, R. G. Leigh, C. Marteau and P. M. Petropoulos,Carroll Structures, Null Geometry and Conformal Isometries, Phys. Rev. D100(2019) no.4, 046010, doi:10.1103/PhysRevD.100.046010 [arXiv:1905.02221 [hep-th]]

arXiv 2019

[5] [3]

Fontanella,Non-extremal near-horizon geometries, Class

A. Fontanella,Non-extremal near-horizon geometries, Class. Quant. Grav.40(2023) no.13, 135006, doi:10.1088/1361-6382/acd980 [arXiv:2211.03861 [gr-qc]]

arXiv 2023

[6] [4]

Lambert and J

N. Lambert and J. Smith,Non-relativistic M2-branes and the AdS/CFT correspon- dence, JHEP06(2024), 009, doi:10.1007/JHEP06(2024)009. [arXiv:2401.14955 [hep- th]]

arXiv 2024

[7] [5]

Fontanella and J

A. Fontanella and J. M. Nieto Garc´ ıa,Constructing nonrelativistic AdS5/CFT4 holog- raphy, Phys. Rev. D111(2025) no.2, 026003, [erratum: Phys. Rev. D112(2025) no.2, 029902], doi:10.1103/PhysRevD.111.026003, [arXiv:2403.02379 [hep-th]]

arXiv 2025

[8] [6]

Lambert and J

N. Lambert and J. Smith,Non-relativistic intersecting branes, Newton-Cartan geometry and AdS/CFT, JHEP07(2024), 224, doi:10.1007/JHEP07(2024)224, [arXiv:2405.06552 [hep-th]]

arXiv 2024