Gauge coupling as a consequence of a multi-valued phase

3 Gauge coupling as a consequence of a multi-valued phase

In this section we study the consequences of the multi-valuedness [[42], [70], [13]] of the quantity $˜S (x, t)$ in the present theory. We assume that $S˜(x, t)$ may be written as a sum of a single-valued part $S (x, t )$ and a multi-valued part $N˜$ . Then, given that (5) holds, the derivatives of $S˜ (x, t )$ may be written in the form
$∂ S˜ ∂ S ∂ S˜ ∂ S e ----- = ----- + eΦ, ------- = -------- --A , k ∂ t ∂ t ∂ xk ∂ xk c$ (16)

where the four functions $Φ$ and $A k$ are proportional to the derivatives of $N˜$ with respect to $t$ and $x k$ respectively (Note the change in sign of $Φ$ and $A k$ in comparison to [32]; this is due to the fact that the multi-valued phase is now denoted by $S˜$ ). The physical motivations for introducing the pre-factors $e$ and $c$ in Eq. (16) have been extensively discussed elsewhere, see [28], [32], in an electrodynamical context. In agreement with Eq. (16), $S˜$ may be written in the form [[28], [32]]
$∫ e x,t [ ] ˜ -- ′ ′ ′ ′ ′ ′ S (x, t; C ) = S (x, t )- dx kAk (x , t ) - cdt ϕ (x , t ) , c x0,t0;C$ (17)

as a path-integral performed along an arbitrary path $C$ in four-dimensional space; the multi-valuedness of $˜ S$ simply means that it depends not only on $x, t$ but also on the path $C$ connecting the points $x , t 0 0$ and $x, t$ .

The quantity $˜S$ cannot be a physical observable because of its multi-valuedness. The fundamental physical quantities to be determined by our (future) theory are the four derivatives of $S˜$ which will be rewritten here as two observable fields $- E˜ (x, t)$ , $˜pk (x, t)$ ,

∂ S (x, t) - E˜ (x, t ) = ------------- + eΦ (x, t ), (18) ∂ t ∂ S (x, t) e ˜p (x, t ) = ------------- - --A (x, t ), (19) k k ∂ xk c

with dimensions of energy and momentum respectively.

We encounter a somewhat unusual situation in Eqs. (18), (19): On the one hand the left hand sides are observables of our theory, on the other hand we cannot solve our (future) differential equations for these quantities because of the peculiar multi-valued structure of $˜ S$ . We have to use instead the decompositions as given by the right hand sides of (18) and (19). The latter eight terms (the four derivatives of $S$ and the four scalar functions $Φ$ and $A k$ ) are single-valued (in the mathematical sense) but need not be unique because only the left hand sides are uniquely determined by the physical situation. We tentatively assume that the fields $Φ$ and $Ak$ are ’given’ quantities in the sense that they represent an external influence (of ’external forces’) on the considered statistical situation. An actual calculation has to be performed in such a way that fixed fields $Φ$ and $Ak$ are chosen and then the differential equations are solved for $S$ (and $ρ$ ). However, as mentioned already, what is actually uniquely determined by the physical situation is the sum of the two terms on the right hand sides of (18) and (19). Consequently, a different set of fixed fields $′ Φ$ and $′ A k$ may lead to a physically equivalent, but mathematically different, solution $′ S$ in such a way that the sum of the new terms [on the right hand sides of (18) and (19)] is the same as the sum of the old terms. We assume here, that the formalism restores the values of the physically relevant terms. This implies that the relation between the old and new terms is given by

′ S (x, t) = S (x, t) + φ (x, t ) (20) ′ 1-∂-φ--(x,--t-) Φ (x, t) = Φ (x, t) - (21) e ∂ t ′ c-∂-φ--(x,--t-) A k (x, t) = Ak (x, t ) + , (22) e ∂ xk

where $φ (x, t)$ is an arbitrary, single-valued function of $xk , t$ . Consequently, all ’theories’ (differential equations for $S$ and $ρ$ defined by the assumptions listed in section 2) should be form-invariant under the transformations (20)-(22). These invariance transformations, predicted here from general considerations, are (using an arbitrary function $χ = cφ ∕e$ instead of $φ$ ) denoted as ’gauge transformations of the second kind’.

The fields $Φ (x, t)$ and $A (x, t ) k$ describe an external influence but their numerical value is undefined; their value at $x, t$ may be changed according to (21) and (22) without changing their physical effect. Thus, these fields cannot play a local role in space and time like forces and fields in classical mechanics and electrodynamics. What, then, is the physical meaning of these fields ? An explanation which seems obvious in the present context is the following: They describe the statistical effect of an external influence on the considered system (ensemble of identically prepared individual particles). The statistical effect of a force field on an ensemble may obviously differ from the local effect of the same force field on individual particles; thus the very existence of fields $Φ$ and $Ak$ different from $⃗ E$ and $⃗ B$ is no surprise. The second common problem with the interpretation of the ’potentials’ $Φ$ and $Ak$ is their non-uniqueness. It is hard to understand that a quantity ruling the behavior of individual particles should not be uniquely defined. In contrast, this non-uniqueness is much easier to accept if $Φ$ and $A k$ rule the behavior of ensembles instead of individual particles. We have no problem to accept the fact that a function that represents a global (integral) effect may have many different local realizations.

It seems that this interpretation of the potentials $Φ$ and $Ak$ is highly relevant for the interpretation of the effect found by [1]. If QT is interpreted as a theory about individual particles, the Aharonov-Bohm effects imply that a charged particle may be influenced in a nonlocal way by electromagnetic fields in inaccessible regions. This paradoxical prediction, which is however in strict agreement with QT, led even to a discussion about the reality of these effects, see [11], [58], [31], [50]. A statistical interpretation of the potentials has apparently never been suggested, neither in the vast literature about the Aharonov-Bohm effect nor in papers promoting the statistical interpretation of QT; most physicists discuss this nonlocal ’paradox’ from the point of view of ’the wave function of a single electron’. Further comments on this point may be found in section 11.

The expectation value $------------------- (e ) F (x, p, t ) k$ on the right hand side of (14) is to be calculated using local, macroscopic forces whose functional form is still unknown. Both the potentials and these local forces represent an external influence, and it is reasonable to assume that the (nonlocal) potentials are the statistical representatives of the local forces on the r.h.s. of Eq. (14). The latter have to be determined by the potentials but must be uniquely defined at each space-time point. The gauge-invariant fields
$1- ∂-Ak---- -∂-Φ--- ∂--Aj-- Ek = - - , Bk = ϵkij , c ∂ t ∂ xk ∂ xi$ (23)

fulfill these requirements. As a consequence of the defining relations (23) they obey automatically the homogeneous Maxwell equations.

In a next step we rewrite the second term on the l.h.s. of Eq. (14). The commutator terms (6) take the form
$( ) ( ) 1 ∂ A ∂ Φ e ∂ A ∂ A ˜ -------k-- ------- ˜ -- -----j- -----k-- S [0,k ] = - e + , S [j,k ] = - . c ∂ t ∂ xk c ∂ xk ∂ xj$ (24)

As a consequence, they may be expressed in terms of the local fields (23), which have been introduced above for reasons of gauge-invariance. Using (24), (23) and the relation (19) for the momentum field, Eq. (14) takes the form
$⌊ ⌋ ∫ ( )2 ∞ ˜ ∑ ˜ 3 -∂-ρ---⌈ ∂-S-- -1---- -∂-S--- ⌉ - d x + + V - ∞ ∂ xk ∂ t 2m ∂ xj j , ∫ ∞ [ ] ------------------- 3 e (e ) + d x ρ --ϵ ˜viBj + eE = F (x, p, t) c kij k k - ∞$ (25)

with a velocity field defined by $v˜i = p˜i ∕m$ . Thus, the new terms on the l.h.s. of (25) - stemming from the multi-valuedness of $S˜$ - take the form of an expectation value (with $R3$ as sample space) of the Lorentz force field
$e ⃗F (x, t) = e ⃗E (x, t) + --⃗˜v (x, t) × ⃗B (x, t), L c$ (26)

if the particle velocity is identified with the velocity field $⃗ ˜v (x, t)$ .

The above steps imply a relation between potentials and local fields. From the present statistical (nonlocal) point of view the potentials are more fundamental than the local fields. In contrast, considered from the point of view of macroscopic physics, the local fields are the physical quantities of primary importance and the potentials may (or may not) be introduced for mathematical convenience.

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