Abstract
The standard \(\mathbb{U}(1)\) “gauge principle” or “gauge argument” produces an exact potential A=dλ and a vanishing field F=d 2 λ=0. Weyl (in Z. Phys. 56:330–352, 1929; Rice Inst. Pam. 16:280–295, 1929) has his own gauge argument, which is sketchy, archaic and hard to follow; but at least it produces an inexact potential A and a nonvanishing field F=dA≠0. I attempt a reconstruction.
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Notes
It is sometimes known as \(\mathrm{Spin}^{\mathbb{C}}\); see Friedrich [5] §1.6 for instance.
Weyl [25], p. 348: “§6. Elektrisches Feld. Wir kommen jetzt zu dem kritischen Teil der Theorie. Meiner Meinung nach liegt der Ursprung und die Notwendigkeit des elektromagnetischen Feldes in folgendem begründet.”
Göckeler & Schücker [6], p. 48: “In physical terms we may interpret the requirement of local gauge invariance (independence of the fields at different spacetime points) as expressing the absence of (instantaneous) action at a distance.” Ryder [19], p. 93: “So when we perform a rotation in the internal space of φ at one point, through an angle Λ, we must perform the same rotation at all other points at the same time. If we take this physical interpretation seriously, we see that it is impossible to fulfil, since it contradicts the letter and spirit of relativity, according to which there must be a minimum time delay equal to the time of light travel. To get round this problem we simply abandon the requirement that Λ is a constant, and write it as an arbitrary function of space-time, Λ(x μ). This is called a ‘local’ gauge transformation, since it clearly differs from point to point.” Teller [23], p. S469: “why should we expect invariance under a local phase transformation to begin with? The plausibility of such invariance probably arises with a misleading analogy with global phase transformations which can be imposed on individual state functions with no change of description.” See also Sakurai [20], p. 16, Aitchison & Hey [1], p. 176, Mandl & Shaw [13], p. 263, Ramond [18], pp. 183–191, O’Raifeartaigh [16], p. 118. One is reminded of Weyl’s rejection ([25], p. 331; [26], p. 286) of distant parallelism.
My analysis owes much to Lyre [9–12]. But
$$\langle\varphi|P|\varphi\rangle=\langle\varphi U|UPU^\dagger|U\varphi \rangle \neq\langle\varphi U|P|U\varphi\rangle $$seems relevant to his claim [12, pp. 649–651], that local phase transformations are not observable. I would say they are—unless one compensates to restore invariance. P. 651 he writes that: “local phase transformations are already unmasked as not observable. From this insight, however, the whole logic of the received view breaks down. Since the introduction of an interaction field as intended by the received view seemingly changes physics (those fields are even directly observable themselves), it is necessary from this view to consider local gauge transformations as changing physics as well in order to tell the story about compensation. Since, however, local gauge transformations can be shown as not observable, the received view proves itself untenable.” It is untenable because the added term dλ is exact. But even if dλ is electromagnetically unobservable, it is quantum-mechanically observable: 〈φ|P|φ〉≠〈φ|P λ |φ〉.
Ryder [19], p. 95: “the electromagnetic field arises naturally by demanding invariance of the action […] under local (x-dependent) rotations […].”
Auyang [2], p. 58, Brown [4], pp. 50–53, Teller [23], pp. S468–S469, Lyre [9–12], Healey [7], p. 438, Martin [14], p. S229, Martin [15], p. 45. But the general structure of the covariant derivative is about right; Lyre [10], p. 84: “Denn wenngleich das Eichprinzip […] nicht zwingend auf nichtflache Konnektionen führt, so ist ja doch die in der kovarianten Ableitung vorgegebene Struktur des Wechselwirkungterms auch für den empirisch bedeutsamen Fall nicht-verschwindender Feldstärken korrekt beschrieben. Diese Wechselwirkungsstruktur is also tatsächlich aus der lokalen Eichsymmetrie-Forderung hergeleitet.”
Cf. Weyl [26], p. 283.
One can wonder what the gauge argument is for if the inexact potential A was already there to begin with. The exact term added in (3) has more to do with the invariance of F=dA=dA′ than with the gauge argument.
Weyl [25], p. 333: “man beschränke sich auf solche lineare Transformationen U von ψ 1, ψ 2, deren Determinante den absoluten Betrag 1 hat.”
Weyl [26], p. 291: “It is my firm conviction that we must seek the origin of the electromagnetic field in another direction. We have already mentioned that it is impossible to connect the transformations of the ψ in a unique manner with the rotations of the axis system; however we may attempt to accomplish this by means of invariants which can be used as constituents of an action quantity we always find that there remains an arbitrary “gauge factor” e iλ. Hence the local axis-system does not determine the components of ψ uniquely, but only within such a factor of absolute magnitude 1.” Weyl [27], p. 195: “Aus der Natur, dem Transformationsgesetz der Größe ψ ergibt sich, daß die vier Komponenten ψ ϱ relativ zum lokalen Achsenkreuz nur bis auf einen gemeinsamen Proportionalitätsfaktor e iλ durch den physikalischen Zustand bestimmt sind, dessen Exponent λ willkürlich vom Orte in Raum und Zeit abhängt, und daß infolgedessen zur eindeutigen Festlegung des kovarianten Differentials von ψ eine Linearform ∑ α f α dx α erforderlich ist, die so mit dem Eichfaktor in ψ gekoppelt ist, wie es das Prinzip der Eichinvarianz verlangt.”
Weyl [25], p. 348: “In der speziellen Relativitätstheorie muß man diesen Eichfaktor als eine Konstante ansehen, weil wir hier ein einziges, nicht an einen Punkt gebundes Achsenkreuz haben.” Weyl [26], p. 291: “In the special theory of relativity, in which the axis system is not tied up to any particular point, this factor is a constant.”
The gauge groups become infinite-dimensional. Weyl [25], p. 348: “Anders in der allgemeinen Relativitätstheorie: jeder Punkt hat sein eigenes Achsenkreuz und darum auch seinen eigenen willkürlichen Eichfaktor; dadurch, daß man die starre Bindung der Achsenkreuze in verschiedenen Punkten aufhebt, wird der Eichfaktor notwendig zu einer willkürlichen Ortsfunktion.” Weyl [26], p. 291: “But it is otherwise in the general theory of relativity when we remove the restriction binding the local axis-systems to each other; we cannot avoid allowing the gauge factor to depend arbitrarily on position.”
Here I am indebted to Johannes Huisman.
Weyl [25], p. 348: “Dann ist aber auch die infinitesimale lineare Transformation dE der ψ, welche der infinitesimalen Drehung dγ entspricht, nicht vollständig festgelegt, sondern dE kann um ein beliebiges rein imaginäres Multiplum i⋅df der Einheitsmatrix vermehrt werden.” Weyl [26], p. 291: “Then there remains in the infinitesimal linear transformation dE of ψ, which corresponds to the given infinitesimal rotation of the axis-system, an arbitrary additive term +idφ⋅1.”
Weyl [25], p. 348: “Zur eindeutigen Festlegung des kovarianten Differentials δψ von ψ hat man also außer der Metrik in der Umgebung des Punktes P auch ein solches df für jedes von P ausgehende Linienelement \(\overrightarrow{PP}'=(dx)\) nötig. Damit δψ nach wie vor linear von dx abhängt, muß
$$df=f_p(dx)^p $$eine Linearform in den Komponenten des Linienelements sein. Ersetzt man ψ durch e iλ, so muß man sogleich, wie aus der Formel für das kovariante Differential hervorgeht, df ersetzen durch df−dλ.” Weyl [26], p. 291: “The complete determination of the covariant differential δψ of ψ requires that such a dφ be given. But it must depend linearly on the displacement PP′: dφ=φ p (dx)p, if δψ shall depend linearly on the displacement. On altering ψ by multiplying it by the gauge factor e iλ we must at the same time replace dφ by dφ−dλ as is immediately seen from this formula of the covariant differential.” Weyl’s notation is confusing: whereas the one-form dλ (which is a differential) is necessarily exact, df and dφ (my A) aren’t.
∇⋅B=0 and ∇×E+∂ t B=0.
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Afriat, A. Weyl’s Gauge Argument. Found Phys 43, 699–705 (2013). https://doi.org/10.1007/s10701-013-9712-x
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DOI: https://doi.org/10.1007/s10701-013-9712-x