Normal frames and linear transports along paths in line bundles. Applications to classical electrodynamics

Bozhidar Z. Iliev (Institute for Nuclear Research; Bulgaria); Bulgarian Academy of Sciences; Nuclear Energy; Sofia

REVIEW

Normal frames and linear transports along paths in line bundles. Applications to classical electrodynamics

Bozhidar Z. Iliev (Institute for Nuclear Research, Nuclear Energy, Bulgarian Academy of Sciences, Sofia, Bulgaria)

Mathematical Physics · Submitted 1 Mar 2006 · 24 references

Pith's verdict: A competent but largely review-level reformulation of electrodynamics as a line-bundle transport; the advertised inertial-frame criterion fails on non-simply-connected regions and (6.1) has a sign error.

T0 · AI review

How this review was made: An AI referee (deepseek-v4-flash) read the paper on August 28, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 2 major objections · 4 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv IDmath-ph/0603002 (version 2)

Pith IDLLDAVLMU, Pith's permanent record of this paper and its review

CitationBozhidar Z. Iliev (Institute for Nuclear Research, Bulgaria), Bulgarian Academy of Sciences, Nuclear Energy, Sofia. "Normal frames and linear transports along paths in line bundles. Applications to classical electrodynamics." arXiv:math-ph/0603002 (2006). https://pith.science/paper/LLDAVLMU

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

Dear colleague,

The one-sentence version: this is a clean but mostly review-level reformulation of classical electrodynamics as a transport in a line bundle, and its advertised inertial-frame criterion breaks on non-simply-connected regions. If you need a pedagogical route from path transports to gauge transformations, it is useful; if you are looking for new physics, there is none, and the author says so.

What is genuinely good: the expository engine in Sections 2-4 is careful. The line-bundle specialization makes frame-change formulas transparent, and the identification of a gauge transformation with a change of fibre frame (lambda through ln a) is presented clearly. The table of gauge conditions is a nice touch. The paper is intellectually honest: it repeatedly states that the transport description is equivalent to the standard connection picture.

Now the soft spots, in rough order of severity.

First, Theorem 3.2 is quoted from [1] without proof and, as stated, is false. Vanishing R means the connection 1-form is closed, not exact. A normal frame requires solving d ln a = -omega, which has a global solution only when omega is exact. On S^1 with constant coefficient, R = 0 but holonomy is nontrivial, so no global normal frame exists. The paper's own Aharonov-Bohm paragraph is exactly this situation, so Section 6's 'frames inertial on U iff F|_U = 0' needs U simply connected, or with trivial holonomy. That is a load-bearing fix, not a nitpick.

Second, equation (6.1) has a sign error. With the paper's own convention, setting Gamma_mu = A_mu into (3.5) gives R_mu_nu(A) = F_mu_nu, not R_mu_nu(-A).

Third, the asserted equivalence to standard gauge theory is true but never demonstrated; a referee would want it written out, including sign conventions.

The reference list is dominated by the author's earlier work; that is defensible since the formalism is his, but external anchors would help.

Bottom line: this is a readable, honest exposition of a known equivalence. The central inertial-frame claim is false without a topological hypothesis, and the sign error suggests the manuscript has not had a careful pass. I would not send it to peer review as is; it could become acceptable for a teaching-oriented venue after those two fixes.

Referee report

Significance

If the central claims are correct, the paper provides a reformulation of classical electrodynamics in the language of linear transports along paths, a framework that is somewhat more general than the usual connection-based formalism. The identification of normal frames with inertial frames gives a concrete geometric meaning to gauge choices, and the Aharonov–Bohm discussion is physically suggestive. The paper is largely a review of the author's previous work, but the specialization to line bundles and the electrodynamics application make it potentially useful as a self-contained introduction. However, the main theorem on which the inertial-frame characterization rests is stated without a necessary topological hypothesis, and the asserted equivalence with standard gauge theory is not demonstrated. Both issues are fixable, but they affect the generality and precision of the paper's central claims.

Major objections (2)

[§3, Theorem 3.2 and §6] Theorem 3.2 states that a C^2 linear transport is Euclidean on a neighborhood U iff the curvature matrices R_μν(-Γ_1,...,-Γ_dim M) vanish on U (eqs. (3.4)–(3.5)). For line bundles, this 'if' direction requires solving d ln a = -Γ_μ dx^μ on U, i.e. the 1-form ω = Γ_μ dx^μ must be exact on U, not merely closed. On a non-simply-connected U a closed but non-exact 1-form gives R=0 yet no global normal frame. A concrete counterexample is M=S^1 with the constant coefficient Γ_θ=α, α∉Z: then R=0 but the holonomy is e^{-2π i α}≠1, and no single-valued nonzero frame a(θ) can satisfy d ln a = -α dθ. The paper's own Aharonov–Bohm paragraph in §6 acknowledges exactly this situation, stating that when L is path-dependent on U (with F=0), no frames normal on U exist. Thus the sentence in §6 that frames normal on U exist iff F|_U=0, and the subsequent identification of inertial frames with such frames, requires the additional hypothesis that U is simply connected or, equivalently, that the transport has trivial holonomy on U. Without this hypothesis the inertial-frame characterization is false.
[§5, 'With a little effort, one can show...'] The paper asserts that the proposed linear-transport description of the electromagnetic field is equivalent to the standard connection-based gauge theory, but no proof or precise statement of this equivalence is given. This claim is central to the physical interpretation: it justifies calling the transport coefficients 'potentials' and the curvature 'field strength'. The author should state the precise correspondence between the transport coefficients Γ_μ and a connection 1-form, and prove that the parallel transport generated by that connection reproduces the transport L, at least locally. Alternatively, the claim should be weakened to an explicit local statement, since the global equivalence may require additional hypotheses related to the topological issues raised above.

Minor objections (4)

[Eq. (3.7)] In the definition of R^N_{αβ}, the second term appears as -∂Γ_β/∂x^α, whereas (3.5) has +∂Γ_ν/∂x^μ. This looks like a sign typo; please correct it.
[Eqs. (3.5) and (6.1)] The notation R_μν(-Γ_1,...,-Γ_dim M) is potentially confusing. If R_μν is meant to be a function of its arguments, substituting -Γ into the expression in (3.5) would change the sign of the derivative terms, making (6.1) inconsistent with (5.1). Please clarify the intended convention, e.g. by defining R_μν(Y) explicitly.
[§1, Introduction] There is a repeated word in 'examples of such frames are the the frames (and possibly coordinates)'; please remove the duplication.
[§7, Conclusion] The phrase 'without selﬁntersections' contains a typo (missing space); it should read 'without self-intersections'.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

An electromagnetic field can be identified with a linear transport along paths in a line bundle over spacetime, so that inertial frames are exactly the frames in which the four-potential vanishes.

The reading

This paper argues that the classical electromagnetic field can be viewed not as a connection but as a linear transport along paths in a one-dimensional vector bundle over spacetime, with the four-potential A playing the role of the transport's coefficients. In this picture, the field strength F = dA becomes the curvature of the transport, and a gauge transformation is simply a change of frame in the bundle's fibers. The payoff is a sharp geometric criterion for inertial frames: on a region U, a frame in which the potentials vanish exists exactly when the transport is path-independent on U, which (by the cited Theorem 3.2) happens exactly when F vanishes on U. The paper also notes that normal frames, and hence inertial frames, always exist at single points and along non-self-intersecting paths, but exist on higher-dimensional regions only for special field configurations.

What carries the argument

The central object is a linear transport along paths: a family of linear maps L_{s->t} between fibers over the endpoints of a path gamma, with composition L_{r->t} = L_{s->t} composed with L_{r->s}. In a line bundle its matrix is a single number L(t,s;gamma)=f(s;gamma)/f(t;gamma), equivalently an exponential of minus the integral of a coefficient Gamma(s;gamma)=A_mu(gamma(s)) gamma-dot^mu(s). The paper identifies the potential A_mu with the 3-index coefficients Gamma_mu of the transport, so that the curvature identity F_mu_nu = R_mu_nu(-A) holds, and the 'normal frame' condition reduces to A'_mu = 0 on U.

Extended reading notes

Core claim

The central claim is that an electromagnetic field can be identified with a linear transport along paths in a one-dimensional real vector bundle over the spacetime manifold, with the potential A_mu identified with the 3-index coefficients Gamma_mu of the transport. With this identification, the field-strength tensor F_mu_nu = partial_nu A_mu - partial_mu A_nu coincides exactly with the curvature R_mu_nu defined from the transport coefficients, and the gauge transformation A -> A + d $\lambda$ becomes the frame change e -> $e^{{lambda}}$ e in the bundle. Therefore a frame normal on U (one in which the transport's matrix is the identity along every path in U) is precisely a frame in which the potentials vanish on U; the paper calls these inertial frames for the electromagnetic field. Because normal frames always exist at a point and along non-self-intersecting paths, the paper concludes that inertial frames for electrodynamics always exist in those restricted senses, and that the equivalence principle for electromagnetism is a provable theorem; on neighborhoods they exist exactly when the field strength vanishes there.

Load-bearing premise

The conclusion rests on Theorem 3.2, quoted from the author's earlier work without proof here, that a transport admits normal frames on a neighborhood exactly when its curvature matrices vanish there; if that theorem or the claimed equivalence to connection-based gauge theory fails, the identification of normal frames with inertial frames for electrodynamics does not follow.

Editorial extensions

If this is right

In this picture, the gauge transformation A -> A + d lambda is not an extra symmetry but a change of the bundle frame e -> e^{lambda} e, so gauge freedom is built into the geometry.
A frame inertial on a region U exists exactly when the field strength vanishes on U (for neighborhoods); physically, an inertial frame for the electromagnetic field is one in which the potentials vanish.
Normal, hence inertial, frames always exist at a single spacetime point and along every non-self-intersecting path, so an equivalence principle for electrodynamics holds in those restricted senses.
If the transport is path-dependent on U, no normal frame on U exists, giving a classical analogue of the quantum effect where potentials act even when the local field strength is zero.

Reading between the lines

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

A direct corollary not spelled out: on non-contractible regions, F = 0 is not enough for normal frames, since path-independence is the operative condition; this ties the paper's criterion to the holonomy of the transport.
The formalism suggests a testable reformulation of gauge fixing: every gauge condition (Lorenz, Coulomb, axial, etc.) is a partial choice of a frame e^{lambda} in the line bundle; one could enumerate which conditions correspond to geometric normal-frame data.
If the identification is right, minimal coupling could be derived from the equivalence principle by requiring the interacting field's bundle to be the line bundle of the electromagnetic transport, as the paper hints at the end.

Original abstract

The definitions and some basic properties of the linear transports along paths in vector bundles and the normal frames for them are recalled. The formalism is specified on line bundles and applied to a geometrical description of the classical electrodynamics. The inertial frames for this theory are discussed.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,838 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.05 · human participants 0.02 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.02 · compares with prior results 0.06 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

The definitions and some basic properties of the linear transports along paths in vector bundles and the normal frames for them are recalled. The formalism is specified on line bundles and applied to a geometrical description of the classical electrodynamics. The inertial frames for this theory are discussed.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The definitions and some basic properties of the linear transports along paths in vector bundles and the normal frames for them are recalled.	background 0.98	0.11	0.04	asserted 0.91
s1	The formalism is specified on line bundles and applied to a geometrical description of the classical electrodynamics.	contribution 0.77	0.82	0.20	asserted 0.99
s2	The inertial frames for this theory are discussed.	contribution 0.98	0.60	0.11	unclear 0.53

What kind of paper the classifier reports this is: theoretical result 0.67 · evidence from proof or derivation 0.93.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (24)

[1]

Bozhidar Z. Iliev. Normal frames and linear transports a long paths in vector bundles. JP Journal of Geometry and Topology , 5(3):187–250, 2005. http://arXiv.org e-Print archive, E-print No. gr-qc/9809 084, September 1998 (last revision: March 2005)

View on Pith 2005
[2]

Bozhidar Z. Iliev. Fibre bundle formulation of nonrelat ivistic quantum mechanics (full version). http://arXiv.org e-Print archive, E-print No. quant-ph/0 004041, April 2000

View on Pith 2000
[3]

Bozhidar Z. Iliev. Normal frames for derivations and lin ear connections and the equivalence prin- ciple. Journal of Geometry and Physics , 45(1–2):24–53, February 2003. DOI No. 10.1016/S0393- 0440(02)00119-5 http://arXiv.org e-Print archive, E-print No. hep-th/011 0194, October 2001

View on Pith doi:10.1016/s0393- 2003
[4]

Bozhidar Z. Iliev. Is the principle of equivalence a prin ciple? Journal of Geometry and Physics , 24(3):209–222, 1998. DOI No. 10.1016/S0393-0440(97)0001 1-9 http://arXiv.org e-Print archive, E-print No. gr-qc/9806 062, June 1998

View on Pith doi:10.1016/s0393-0440(97)0001 1998
[5]

Bozhidar Z. Iliev. Equivalence principle in classical e lectrodynamics. In Andrew Chubykalo, Vladimir Onoochin, Augusto Espinoza, and Roman Smirnov-Ru eda, editors, Has the last word been said on classical electrodynamics? – New Horizons , pages 385–403. Rinton Press, Princeton, NJ, 2004. http://arXiv.org e-Print archive, E-print No. gr-qc/0303 002, March 2003

View on Pith 2004
[6]

Walter A. Poor. Diﬀerential geometric structures . McGraw-Hill Book Company Inc., New York, 1981

View on Pith 1981
[7]

W. Greub, S. Halperin, and R. Vanstone. De Rham cohomology of manifolds and vector bundles , volume 1 of Connections, Curvature, and Cohomology . Academic Press, New York and London, 1972

View on Pith 1972
[8]

N. P. Konopleva and V. N. Popov. Gauge ﬁelds . Hardwood Academic Publishers, Chur-London- New York, second edition, 1981. Translation from Russian: A tomizdat, Moscow, 1972 (1 ed.), 1980 (2 ed.)

View on Pith 1981
[9]

W. Drechsler and M. E. Mayer. Fibre bundle techniques in gauge theories , volume 67 of Lecture notes in physics . Springer-Verlag, Berlin-Heidelberg-New York, 1977

View on Pith 1977
[10]

L. D. Landau and E. M. Lifshitz. Classical theory of ﬁelds , volume II of Course of theoretical physics. Pergamon Press, Oxford, 5 edition, 1967. Translation from Russian, Nauka, Moscow, 1973. Bozhidar Z. Iliev: Normal frames and transports in line bundles. Electrodynamics 11

View on Pith 1967
[11]

C. Nash and S. Sen. Topology and Geometry for physicists . Academic Press, London-New York, 1983

View on Pith 1983
[12]

John Baez and Javier P. Muniain. Gauge ﬁelds, knots and gravity , volume 4 of Series in knots and Everything . World Scientiﬁc, Singapore-New Jersey-London-Hong Kong , 1994

View on Pith 1994
[13]

Ludwig Valentin Lorenz. ¨Uber die Intensit¨ at der Schwingungen des Lichts mit den ele ktrischen Str¨ omen.Annalen der Physik und Chemie , 131:243–263, 1867

View on Pith
[14]

Edmund Whittaker. A history of the theories of aether and electricity , volume 1. The classical theories. of Harper torchbooks / The science library . Harper & brothers, New York, 1960. Origi- nally published by Thomas Nelson & Son Ltd, London, 1910; rev ised and enlarged 1951. See also the 1989 edition: New York: Dover, 1989

View on Pith 1960
[15]

Paul Roman. Introduction to quantum ﬁeld theory . John Wiley&Sons, Inc., New York-London- Sydney-Toronto, 1969

View on Pith 1969
[16]

M. G¨ ockeler and T. Sch¨ ucker. Diﬀerential geometry, gauge theories, and gravity . Cambridge Univ. Press, Cambridge, 1987

View on Pith 1987
[17]

A. A. Slavnov and L. D. Fadeev. Introduction to quantum theory of gauge ﬁelds . Nauka, Moscow,

View on Pith
[18]

Y. Aharonov and D. Bohm. Signiﬁcance of electromagneti c potentials in quantum theory. Phys. Rev., 115(3):485, 1959

View on Pith 1959
[19]

H. J. Bernstein and A. V. Phillips. Fibre bundles and qua ntum theory. Scientiﬁc America , 245:94–104, July 1981. Reprinted: Uspekhi Fizicheskikh Na uk, vol. 136, No. 4, pp. 665–692, 1982, In Russian

View on Pith 1981
[20]

Bozhidar Z. Iliev. Normal frames and the validity of the equivalence principle: I. Cases in a neighborhood and at a point. Journal of Physics A: Mathematical and General , 29(21):6895– 6901, 1996. DOI No. 10.1088/0305-4470/29/21/020 http://arXiv.org e-Print archive, E-print No. gr-qc/9608 019, August 1998

View on Pith doi:10.1088/0305-4470/29/21/020 1996
[21]

Bozhidar Z. Iliev. Normal frames and the validity of the equivalence principle: II. The case along paths. Journal of Physics A: Mathematical and General , 30(12):4327–4336, 1997. DOI No.10.1088/0305-4470/30/12/019 http://arXiv.org e-Print archive, E-print No. gr-qc/9709 053, September 1997

View on Pith doi:10.1088/0305-4470/30/12/019 1997
[22]

Bozhidar Z. Iliev. Normal frames and the validity of the equivalence principle: III. The case along smooth maps with separable points of self-intersection. Journal of Physics A: Mathematical and General, 31(4):1287–1296, January 1998. DOI No. 10.1088/0305-447 0/31/4/016 http://arXiv.org e-Print archive, E-print No. gr-qc/9805 088, May 1998

View on Pith doi:10.1088/0305-447 1998
[23]

I. Fatibene, M. Rerraris, and M. Francaviglia. On the ga uge natural structure of modern physics. International journal of geometric methods in modern physic s, 1(4):443–466, 2004

View on Pith 2004
[1978]

In Russian. English translation: Gauge Fields: An Int roduction to Quantum Theory, Addison-Wesley (Frontiers in Physics vol. 83), second edit ion, 1991

View on Pith 1991

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

Bozhidar Z. Iliev. Normal frames and linear transports a long paths in vector bundles. JP Journal of Geometry and Topology , 5(3):187–250, 2005. http://arXiv.org e-Print archive, E-print No. gr-qc/9809 084, September 1998 (last revision: March 2005)

View on Pith 2005

[2] [2]

Bozhidar Z. Iliev. Fibre bundle formulation of nonrelat ivistic quantum mechanics (full version). http://arXiv.org e-Print archive, E-print No. quant-ph/0 004041, April 2000

View on Pith 2000

[3] [3]

Bozhidar Z. Iliev. Normal frames for derivations and lin ear connections and the equivalence prin- ciple. Journal of Geometry and Physics , 45(1–2):24–53, February 2003. DOI No. 10.1016/S0393- 0440(02)00119-5 http://arXiv.org e-Print archive, E-print No. hep-th/011 0194, October 2001

View on Pith doi:10.1016/s0393- 2003

[4] [4]

Bozhidar Z. Iliev. Is the principle of equivalence a prin ciple? Journal of Geometry and Physics , 24(3):209–222, 1998. DOI No. 10.1016/S0393-0440(97)0001 1-9 http://arXiv.org e-Print archive, E-print No. gr-qc/9806 062, June 1998

View on Pith doi:10.1016/s0393-0440(97)0001 1998

[5] [5]

Bozhidar Z. Iliev. Equivalence principle in classical e lectrodynamics. In Andrew Chubykalo, Vladimir Onoochin, Augusto Espinoza, and Roman Smirnov-Ru eda, editors, Has the last word been said on classical electrodynamics? – New Horizons , pages 385–403. Rinton Press, Princeton, NJ, 2004. http://arXiv.org e-Print archive, E-print No. gr-qc/0303 002, March 2003

View on Pith 2004

[6] [6]

Walter A. Poor. Diﬀerential geometric structures . McGraw-Hill Book Company Inc., New York, 1981

View on Pith 1981

[7] [7]

W. Greub, S. Halperin, and R. Vanstone. De Rham cohomology of manifolds and vector bundles , volume 1 of Connections, Curvature, and Cohomology . Academic Press, New York and London, 1972

View on Pith 1972

[8] [8]

N. P. Konopleva and V. N. Popov. Gauge ﬁelds . Hardwood Academic Publishers, Chur-London- New York, second edition, 1981. Translation from Russian: A tomizdat, Moscow, 1972 (1 ed.), 1980 (2 ed.)

View on Pith 1981

[9] [9]

W. Drechsler and M. E. Mayer. Fibre bundle techniques in gauge theories , volume 67 of Lecture notes in physics . Springer-Verlag, Berlin-Heidelberg-New York, 1977

View on Pith 1977

[10] [10]

L. D. Landau and E. M. Lifshitz. Classical theory of ﬁelds , volume II of Course of theoretical physics. Pergamon Press, Oxford, 5 edition, 1967. Translation from Russian, Nauka, Moscow, 1973. Bozhidar Z. Iliev: Normal frames and transports in line bundles. Electrodynamics 11

View on Pith 1967

[11] [11]

C. Nash and S. Sen. Topology and Geometry for physicists . Academic Press, London-New York, 1983

View on Pith 1983

[12] [12]

John Baez and Javier P. Muniain. Gauge ﬁelds, knots and gravity , volume 4 of Series in knots and Everything . World Scientiﬁc, Singapore-New Jersey-London-Hong Kong , 1994

View on Pith 1994

[13] [13]

Ludwig Valentin Lorenz. ¨Uber die Intensit¨ at der Schwingungen des Lichts mit den ele ktrischen Str¨ omen.Annalen der Physik und Chemie , 131:243–263, 1867

View on Pith

[14] [14]

Edmund Whittaker. A history of the theories of aether and electricity , volume 1. The classical theories. of Harper torchbooks / The science library . Harper & brothers, New York, 1960. Origi- nally published by Thomas Nelson & Son Ltd, London, 1910; rev ised and enlarged 1951. See also the 1989 edition: New York: Dover, 1989

View on Pith 1960

[15] [15]

Paul Roman. Introduction to quantum ﬁeld theory . John Wiley&Sons, Inc., New York-London- Sydney-Toronto, 1969

View on Pith 1969

[16] [16]

M. G¨ ockeler and T. Sch¨ ucker. Diﬀerential geometry, gauge theories, and gravity . Cambridge Univ. Press, Cambridge, 1987

View on Pith 1987

[17] [17]

A. A. Slavnov and L. D. Fadeev. Introduction to quantum theory of gauge ﬁelds . Nauka, Moscow,

View on Pith

[18] [18]

Y. Aharonov and D. Bohm. Signiﬁcance of electromagneti c potentials in quantum theory. Phys. Rev., 115(3):485, 1959

View on Pith 1959

[19] [19]

H. J. Bernstein and A. V. Phillips. Fibre bundles and qua ntum theory. Scientiﬁc America , 245:94–104, July 1981. Reprinted: Uspekhi Fizicheskikh Na uk, vol. 136, No. 4, pp. 665–692, 1982, In Russian

View on Pith 1981

[20] [20]

Bozhidar Z. Iliev. Normal frames and the validity of the equivalence principle: I. Cases in a neighborhood and at a point. Journal of Physics A: Mathematical and General , 29(21):6895– 6901, 1996. DOI No. 10.1088/0305-4470/29/21/020 http://arXiv.org e-Print archive, E-print No. gr-qc/9608 019, August 1998

View on Pith doi:10.1088/0305-4470/29/21/020 1996

[21] [21]

Bozhidar Z. Iliev. Normal frames and the validity of the equivalence principle: II. The case along paths. Journal of Physics A: Mathematical and General , 30(12):4327–4336, 1997. DOI No.10.1088/0305-4470/30/12/019 http://arXiv.org e-Print archive, E-print No. gr-qc/9709 053, September 1997

View on Pith doi:10.1088/0305-4470/30/12/019 1997

[22] [22]

Bozhidar Z. Iliev. Normal frames and the validity of the equivalence principle: III. The case along smooth maps with separable points of self-intersection. Journal of Physics A: Mathematical and General, 31(4):1287–1296, January 1998. DOI No. 10.1088/0305-447 0/31/4/016 http://arXiv.org e-Print archive, E-print No. gr-qc/9805 088, May 1998

View on Pith doi:10.1088/0305-447 1998

[23] [23]

I. Fatibene, M. Rerraris, and M. Francaviglia. On the ga uge natural structure of modern physics. International journal of geometric methods in modern physic s, 1(4):443–466, 2004

View on Pith 2004

[24] [1978]

In Russian. English translation: Gauge Fields: An Int roduction to Quantum Theory, Addison-Wesley (Frontiers in Physics vol. 83), second edit ion, 1991

View on Pith 1991