Galilean Covariance Versus Gauge Invariance Germain Rousseaux

Galilean Covariance versus Gauge Invariance Germain Rousseaux

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Germain Rousseaux. Galilean Covariance versus Gauge Invariance. 2009. hal-00440826

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Germain Rousseaux Universitéde Nice-Sophia Antipolis, Laboratoire J.-A. Dieudonné, UMR CNRS-UNS 6621, Parc Valrose, 06108 Nice Cedex 02, France, European Union. (Dated: December 11, 2009) We demonstrate for the first time and unexpectedly that the Principle of Relativity dictates the choice of the ”gauge conditions” in the canonical example of a Gauge Theory namely Classical Electromagnetism. All the known ”gauge conditions” of the literature are interpreted physically as electromagnetic continuity equations hence the ”gauge fields”. The existence of a Galilean Electro- magnetism with TWO dual limits (”electric” and ”magnetic”) is the crux of the problem [1]. A phase-space with the domains of validity of the various ”gauge conditions” is provided and is shown to depend on three characteristic times : the magnetic diffusion time, the charge relaxation time and the transit time of electromagnetic waves in a continuous medium [2].

The Standard Model of Physics is based on the as- torial identities, the fields obey the following constraints B ∂B E sumed existence of a superior principle called Gauge . = 0 and ∂t = . But how are defined the Symmetry which would rule all the laws of Physics: Phys- potentials∇ themselves−∇ ? They × are the mathematical so- ical theories of fundamental significance tend to be gauge lutions of the Maxwell-Minkowski equations written for theories. These are theories in which the physical sys- the excitations: tem being dealt with is described by more variables than ∂D there are physically independent degree of freedom. The .D = ρ and H = + J. (1) physically meaningful degrees of freedom then reemerge as ∇ ∇× ∂t being those invariant under a transformation connecting We have to relate the excitations to the fields thanks the variables (gauge transformation). Thus, one intro- to the constitutive relations for media at rest and then duces extra variables to make the description more trans- the fields to the potentials thanks to their definitions parent and brings in at the same time a gauge symmetry above. The current density features two terms J = to extract the physically relevant content. It is a remark- Jconstitutive + Jexternal. The constitutive current which able occurrence that the road to progress has invariably expresses the matter response to the fields depends on the been towards enlarging the number of variables and intro- medium. For example, in Ohmic conductors, we have ducing a more powerful symmetry rather than conversely ∂A JOhm = σE = σ V whereas in a Supercon- aiming at reducing the number of variables and eliminat- −∇ − ∂t ductor [8], the constitutive relation becomes JSupra = ing the symmetry [3]. Wolfgang Pauli was used to ask at ∗ ∗ ~ne e A the end of tiresome seminars he attended loosely if the m φ ~ . For continuous media at rest the ex- ∇ − principal result presented by the speaker was ”gauge in- citations are related to the fields according to D = ǫE variant” [4]. Hence, the concept of Gauge Theory has and B = µH. We get a system of equations where the emerged progressively in Physics such that the equa- unknowns are the potentials S (A, V ; ǫ, µ, ρ, J) = 0 pro- tions feature variables (”gauge fields”) which are under- vided the sources are given or expressed in function of the determined and in order to remove this degree of liberty potentials which vanish far from the latter or take pre- (”gauge transformations”) a closure assumption (”gauge scribed values on given boundaries. However, the system condition”) is formulated [5]. Similarly, the Principle of S = 0 cannot be solved unless another equation is added. Relativity is known to be a robust safeguard when scaf- This closure assumption is usually known as the ”gauge folding a new theory since the proposed new laws must condition” in the Heaviside-Hertz formulation since the be covariant with respect to the transformations of space- potentials are de facto underdetermined (by the ”gauge ∂f time. transformations” A′ = A + f and V ′ = V [5]) if ∇ − ∂t The goal of this paper is to remove the Gauge sym- and only if they are defined in function of the fields and metry in the most famous example of a supposed Gauge not the reverse as in the Riemann-Lorenz formulation. Theory namely Classical Electromagnetism by revealing In the following, we will show that the closure assump- a conflict with another symmetry that is the Principle of tion is a consequence of the Relativistic or Galilean na- Relativity. To do so, we first emphasize the Riemann- ture of the problem under study. For that purpose, we Lorenz approach to Electromagnetism. Therein the cen- will recall the Stratton ”gauge condition” which is, ac- tral role is played by the vector and scalar potentials cording to us, the most general physical constraint which A and V , unlike the Heaviside-Hertz approach, which can be used all the times. Then, thanks to the Galilean rather relies on the fields B and E themselves (for a jus- limits of Classical Electromagnetism [1, 9, 10, 11, 12], we tification, see [6] and [7]). In this formulation, the fields will approximate the Stratton ”gauge condition” depend- are defined as a function of the potentials (and not the ing on the context and we will recover the other ”gauge B A E ∂A reverse) according to = and = V ∂t . As conditions” introduced in the literature by pointing out a consequence of these definitions∇× and using−∇ obvious− vec- their domain of validity. 2

The Stratton ”gauge condition” was introduced in the mass continuity equation for compressible flows in the Physics at M.I.T. in 1941 by Julius Adams Stratton [13] particular case of the linearized acoustic perturbations. to cope with the propagation of electromagnetic waves As a matter of fact, the mass conservation of a flowing in Ohmic conductors such that the sources are given by fluid is encoded in the following law [17]: ρ = 0 and Jconstitutive = JOhm. Its temporal Fourier ∂ρ transformation was known as early as 1928 by communi- . (ρu)+ =0. (6) cation engineers like John Renshaw Carson from Bell Sys- ∇ ∂t tem [14]. Indeed, from the temporal Fourier transforma- If we perturb the density, pressure and velocity around tion of the Maxwell-Ampère equation Hˆ = iωǫEˆ+σEˆ, ∇× σ a basic state at rest: ρ = ρ0 + δρ, p = p0 + δp and Carson introduced a complex permittivity ǫ = ǫ i ω u = 0 + δu, the continuity equation can be recast in a into the temporal Fourier transformation of the Lorenz− A ∂V Lorenz ”gauge condition” form: ”gauge condition” . + µǫ ∂t = 0 [5, 16] to obtain the temporal Fourier transformation∇ of the Stratton ”gauge ˆ u 1 ∂ δp condition” .A + (iωµǫ + µσ)Vˆ = 0. . (δ )+ 2 = 0 (7) ∇ ∇ cs ∂t ρ0 According to Stratton’s alternative procedure, Gauss’ law .E = 0 implies immediately: 1 ∂p δp ∇ where cs = √ρκ = ∂ρ δρ is the speed of sound q ≃ q 1 2 ∂ analogous to the speed of light in vacuum c = . V + ( .A) = 0 (2) √µ0ǫ0 ∇ ∂t ∇ The Coulomb ”gauge condition” .A = 0 is analogous ∇ which can be solved if and only if the potentials are con- to the mass continuity equation for incompressible flows strained by the Stratton ”gauge condition”: .u = 0 [17] provided that the compressibility (permit- ∇ tivity) vanishes i.e. κ 0 at constant density ρ0 (perme- ∂V ability). As we will see→ later on, this approximation cor- .A + µǫ = µσV. (3) ∇ ∂t − responds to the Galilean (magnetic) limit of the Lorenz ”gauge condition” [9, 10, 11]. In the simple case of constant permeability µ and permit- The Stratton ”gauge condition” is a generalized conti- tivity ǫ, Stratton deduced from the Maxwell-Minkowski’s nuity equation for the vector potential : set the following equations (SStratton = 0):

2 ∂V 2 ∂ V ∂V .A + µǫ = µσV. (8) V µǫ µσ = 0 (4) ∇ ∂t − ∇ − ∂t2 − ∂t The right-hand side is a sink term. The vector potential 2A A is dissipated by Ohmic conduction. Loci of high scalar 2A ∂ ∂ J µǫ 2 µσ = µ external (5) potential are sinks for the vector potential whose flux is ∇ − ∂t − ∂t − directed towards them. The Stratton ”gauge condition” which are the well-known ”telegrapher’s equations”. is analogous to the mass continuity equation with nuclear They were derived previously for the tension and the reactions acting as a sink. current by Vaschy and Heaviside starting from the global Thanks to the above analogy with Fluid Mechanics, it electrical equations of Kirchhoff for circuitry and not di- is now obvious to the reader that the vector (scalar) po- rectly from the local Maxwell-Minkowski equations for tential is a kind of electromagnetic momentum (energy) the fields. As an example, they described the propagation per unit charge [7]. Once again, modern Physics has al- of waves in a coaxial cable with Ohmic dissipation. Later, most completely forgotten the physical meaning of the Paul Poincelot derived its tensorial expression since the potentials as it was formulated by James Clerk Maxwell Stratton ”gauge condition” is not manifestly Relativis- in the nineteenth century and part of his results are redis- tic covariant under the Lorentz transformations of space- covered from time to time either by historians of science time [15]. The more famous Lorenz ”gauge condition” or Physics teachers [7]. [5, 16] is the dissipation-free version of the Stratton’s We have just recalled three examples of ”gauge condi- constraint (σ = 0). As a partial conclusion, it is very tions”. It is clear that the analogy with Fluid Mechan- surprising to notice that the Stratton ”gauge condition” ics advocates for different domains of validity depending is completely absent from modern textbooks and is not on the underlying Physics. Here, we will discuss how even mentioned in the benchmark review paper on the to choose a ”gauge condition” depending on the con- history of Gauge Invariance [5]. text. Our method will be dimensional analysis as often Now, we recall the reader of the physical meaning of in Fluid Mechanics. Our guide will be Relativistic or the potentials [7] and their constraints. As for the Strat- Galilean Covariance. That is why we start by a recap ton ”gauge condition”, the following interpretations of on Galilean Electromagnetism as described by Physicists the ”gauge conditions” are nowhere in modern treat- following Lévy-Leblond and Le Bellac [1, 9, 10, 11, 12] ments of Classical Electromagnetism. The Lorenz ”gauge and Engineers following another M.I.T. researcher James 1 ∂V condition” for vacuum .A + 2 = 0 is analogous to Melcher [2]. ∇ c ∂t 3

1 ǫ L We list first the dimensional quantities. An electro- l = the constitutive length, τem = the light ∗ σ µ cm magnetic phenomenon happens in a spatial arena of ex- q ǫ transit time, τe = σ the charge relaxation time and tension L in a duration τ. The arena is a continuous 2 medium with constitutive properties ǫ, µ and σ taken τm = µσL the magneticp diffusion time such that τem = as constant for simplicity (otherwise they are tensors √τeτm. with time and space dependance). Applying the Vaschy- The Figure 1 displays the different approximations of Buckingham theorem of dimensional analysis [17], we can the Stratton’s constraint depending on the Relativistic construct dimensionless parameters which would char- or Galilean (Magnetic, Electric or Statics) regime for a acterize the electromagnetic response of the continuous given problem. In practice, we compare the magnitude of medium. As we will deal with Galilean approxima- the three terms I, II and III in the Stratton’s constraint L using the scaling laws (i), (ii) or (iii). tions, we introduce v τ the typical velocity of the ≈ 1 system and we compare it with c = the light m √µǫ τm celerity in the continuous medium. The Galilean limit (a) log τ em τ = τm (quasi-static approximation) corresponds to v << cm. If we neglect time dependance in the Stratton system Galilean S 2A Stratton = 0 (∂/∂t = 0 or ∂/∂t 0), we get = Magnetic J 2 ρ ≃ ∇ µ and V = ǫ . In terms of orders of magni- Limit tude− [1, 9, 10,∇ 11, 12]− (the tilde means order of magni- ˜ 2 ˜ ˜ L2 tude), we deduce A L µJ and V ǫ ρ˜. Hence, we construct by hand the≈ dimensionless parameter:≈ τ 1 ˜ Special Galilean log √ǫµ A J˜ τem (9) Relativity Statics ˜ 1 V ≈ ρ˜√ǫµ which characterizes the type of regime [1, 9, 10, 11, 12]: (i) v cm and cmA˜ V˜ Relativistic regime; (ii) Galilean ≃ ≃ → v << cm and cmA˜ >> V˜ (V˜ vA˜) Galilean mag- Electric ≃ → netic limit (magnetoquasi-statics or MQS); (iii) v <

˜ ˜ ∂V A V ∂V ∇.A + µǫ = −µσV .A + µǫ = µσV˜ V (10) ∂t L ∇ τ ∂t − ∇.A ≃ − µσV that is: τ L log ˜ ˜ τem A τ V ∂V L V . + 1 1 = 1 V (11) ∇ A˜ ∂t −l A˜ ∇ A ≃ − σ √µǫ √ǫµ ∗ √ǫµ . µ V ∂V ∇.A + µǫ = −µσV whose mathematical form is simply I + II = III with ∂t the following dimensionless ratios: ∂V ∇.A + µǫ ≃ 0 II v V˜ τem V˜ ∂t = (12) τ = τe I ≈ cm cmA˜ τ cmA˜ FIG. 1: Domains of validity of (a) the Galilean limits and (b) III L V˜ τm V˜ τem V˜ the various ”gauge conditions” in a log-log plot inspired by = = (13) Melcher [2] with dimensionless times as variables. I ≈ l cmA˜ τem cmA˜ τe cmA˜ ∗ Hence, the ”gauge conditions” are continuity equations III τ whose domains of validity depend on the Relativistic or (14) II ≈ τe Galilean nature of the underlying phenomenon and have nothing to do with mathematical closure assumptions and where we introduced the following parameters [2]: taken without physical motivations. 4

According to our results, Gauge Invariance is NOT a ically as electromagnetic energy and momentum per fundamental symmetry of Physics since (1) the ”gauge unit charge; (4) the ”gauge conditions” have domains of transformations” can be avoided by a direct deﬁnition of validity derived from Relativistic or Galilean Covariance. the potentials as mathematical solutions of the Maxwell- Minkowski equations; (2) the ”gauge conditions” are interpreted physically as electromagnetic continuity The author would like to thank Francesca Rapetti for equations; (3) the ”gauge ﬁelds” are interpreted phys- playing the role of a sounding board.

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