Gauge coupling

4 Gauge coupling

At this stage of our study it may be useful to clarify the physical role of the quantities $S˜$ , $S$ , and $A , Φ k$ introduced at the beginning of section 2. The multi-valued function $S˜$ may be represented [17],[19] in the form
$∫ x,t [ ] ˜ e- ′ ′ ′ ′ ′ ′ S (x, t; C ) = S (x, t )- dx kAk (x , t ) - cdt φ (x , t ) , c x ,t ;C 0 0$ (24)

as a path-integral performed along an arbitrary path $C$ in four-dimensional space; the multi-valuedness of $˜ S$ means simply that it depends not only on $x, t$ but also on the path $C$ connecting the points $x0, t0$ 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 $˜ S$ [see (3)] which will be rewritten here using the above notation
∂ ˜S (x, t; C ) ∂ S (x, t)
----------------- = ------------- + eΦ (x, t ),

∂ t ∂ t
˜
∂-S--(x,--t;-C--) ∂-S-(x,---t)- e-
= - Ak (x, t),
∂ xk ∂ xk c
(25)

(26)
But we encounter here a somewhat unusual situation: On the one hand the left hand sides of (25) (26) are the basic physical quantities 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 (25) and (26). The latter terms are single-valued (in the sense of the above definition) 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 $A k$ 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 $A k$ 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 (25) (26). Consequently, a different set of fixed fields $′ Φ$ and $′ A k$ may lead to a different solution $′ S$ in such a way that the sum of the new terms [on the right hand sides of (25) and (26)] 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)

′ 1 ∂ φ (x, t)
---------------
Φ (x, t) = Φ (x, t) -
e ∂ t
′ c ∂ φ (x, t )
---------------
A k (x, t ) = Ak (x, t ) + ,
e ∂ xk
(27)

(28)

(29)
where $φ (x, t)$ is an arbitrary, single-valued function of $x , t k$ . Consequently, all ’theories’ (differential equations for $S$ and $ρ$ defined by the assumptions listed in section 2) will be form-invariant under the transformations (27)-(29). These invariance transformations are (using an arbitrary function $χ = c φ ∕e$ 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 (28) and (29) 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 $A k$ 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 Aharonov-Bohm effect [1], [27]. A statistical interpretation of the potentials has apparently never been suggested, neither in the vast literature on the Aharonov-Bohm effect nor in papers promoting the statistical interpretation of quantum mechanics; most physicists discuss this nonlocal ’paradox’ from the point of view of ”the wave function of a single electron”. Further discussion on the significance of the potentials may be found in section 9.

The expectation value $--------------------- (e ) F (x, p, t ) k$ on the right hand side of (23) 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 plausible to assume that they are not independent from each other. Thus it is reasonable to assume that the (nonlocal) potentials are statistical representations of the same external (local) forces, occurring on the r.h.s. of Eq. (23). These local forces have to be determined by the potentials but must be uniquely defined at each space-time point. The gauge-invariant fields
$1 ∂ Ak ∂ Φ ∂ Aj E = - ---------- - ------, B = ε -------, k c ∂ t ∂ x k kij ∂ x k i$ (30)

fulfill these requirements. As a consequence of the defining relations (30) they obey automatically the homogeneous Maxwell equations.
$∂ ⃗B ∂ 1 ∂ B⃗ ------ ----- ⃗ -------- = 0, × E + = 0. ∂ ⃗r ∂ ⃗r c ∂ t$ (31)

Note that 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.

As a next step we rewrite the second term on the l.h.s. of Eq. (23). The commutator terms (16) take the form
$( ) ( ) 1 ∂ Ak ∂ Φ e ∂ Aj ∂ Ak S˜ = - e ----------+ ------- , S˜ = -- ------- - -------- . [0,k ] c ∂ t ∂ x [j,k ] c ∂ x ∂ x k k j$ (32)

As a consequence, they may be expressed in terms of the local fields (30), which have been introduced above for reasons of gauge-invariance. Using (32),(30) and the relation (4) for the momentum field, Eq. (23) takes the form
$⌊ ( ) ⌋ ∫ ∞ 2 ∂ ρ ∂ S˜ 1 ∑ ∂ ˜S - d3x -------⌈ ----- + ------ ------- + V ⌉ ∂ x ∂ t 2m ∂ x - ∞ k j j , ∫ --------------------- ∞ [ e ] 3 -- (e ) + d x ρ εkij ˜viBj + eEk = F k (x, p, t) - ∞ c$ (33)

with a velocity field $v˜i$ defined by
$˜ 1--∂--S-(x,---t) ˜vi (x, t) = . m ∂ xi$ (34)

Thus, the new terms on the l.h.s. of (33) - stemming from the multi-valuedness of $˜ S$ - take the form of an expectation value (with $3 R$ as sample space) of the Lorentz force field
$e ⃗F (x, t) = e ⃗E (x, t) + --˜ (x⃗, tv) × ⃗B (x, t), L c$ (35)

if the particle velocity is identified with the velocity field (34).

Let us discuss now the nature of the macroscopic forces $(e ) F k (x, p, t )$ entering the expectation value on the r.h.s. of Eq. (33). In our type I parent theory, classical mechanics, there are no constraints for the possible functional form of $(e ) F (x, p, t ) k$ . However, this need not be true in the present statistical framework. As a matter of fact, the way the mechanical potential $V (x, t)$ entered the differential equation for $S$ (in the previous work I) indicates already that such constraints do actually exist. Let us recall that we tacitly restricted the class of forces to those derivable from a potential $V (x, t)$ . If we eliminate this restriction and admit arbitrary forces, with components $F (x, t) k$ , we obtain instead of the above relation (33) the simpler relation [Eq. (24) of I, generalized to three dimensions and arbitrary forces of the form $F (x, t) k$ ]
$⌊ ⌋ ∫ ( ) ∫ ∞ ∂ ρ 1 ∑ ∂ S 2 ∂ S ∞ 3 ------- ⌈ ------ ------- -----⌉ - d x + = dx ρFk (x, t ). - ∞ ∂ xk 2m ∂ xj ∂ t - ∞ j$ (36)

This is a rather complicated integro-differential equation for our variables $ρ (x, t )$ and $S (x, t)$ . We assume now, using mathematical simplicity as a guideline, that Eq. (36) can be written in the common form of a local differential equation. This assumption is of course not evident; in principle the laws of physics could be integro-differential equations or delay differential equations or take an even more complicated mathematical form. Nevertheless, this assumption seems rather weak considering the fact that all fundamental laws of physics take this ’simple’ form. Thus, we postulate that Eq. (36) is equivalent to a differential equation
$( )2 1 ∑ ∂ S ∂ S ------ ------- + ----- + T = 0, 2m ∂ x ∂ t j j$ (37)

where the unknown term $T$ describes the influence of the force $Fk$ but may also contain other contributions. Let us write
$T = - L0 + V ,$ (38)

where $L0$ does not depend on $F k$ , while $V$ depends on it and vanishes for $F → 0 k$ . Inserting (37) and (38) in (36) yields
$∫ ∫ 3 -∂--ρ-- 3 d x ( - L0 + V ) = d x ρFk (x, t). ∂ xk$ (39)

For $F → 0 k$ Eq. (39) leads to the relation
$∫ 3 ∂ ρ d x ------- L = 0, ∂ x 0 k$ (40)

which remains true for finite forces because $L0$ does not depend on $Fk$ . Finally, performing a partial integration, we see that a relation
$∫ ∂ V ∞ F = - ------- + s , d3x ρs = 0, k k k ∂ xk - ∞$ (41)

exists between $Fk$ and $V$ , with a vanishing expectation value of the (statistically irrelevant) functions $s k$ . This example shows that the restriction to gradient fields, made above and in I, is actually not necessary. We may admit force fields which are arbitrary functions of $x$ and $t$ ; the statistical conditions (which play now the role of a ’statistical constraint’) eliminate automatically all forces that cannot be written after statistical averaging as gradient fields.

This is very interesting and indicates the possibility that the present statistical assumptions leading to Schrödinger’s equation may also be responsible, at least partly, for the structure of the real existing (gauge) interactions of nature.

This statistical constraint may also work in the present $p -$ dependent case. We assume that the force in (33) is a standard random variable with the configuration space as sample space (see the discussion in section 4 of I) and that the variable $p$ in $(e ) F (x, p, t ) k$ may consequently be replaced by the field $p˜(x, t)$ [see (4)]. Then, the expectation value on the r.h.s. of (33) takes the form
$--------------------- ∫ ∞ ˜ (e) 3 ∂ S (x, t ) F (x, p, t) = d x ρ (x, t)Hk (x, -------------, t ). k ∂ x - ∞$ (42)

The second term on the l.h.s. of (33) has the same form. Therefore, the latter may be eliminated by writing
$∂ S˜ e 1 ∂ ˜S ∂ S˜ H (x, -----, t) = --ε ----------B + eE + h (x, -----, t), k kij j k k ∂ x c m ∂ xi ∂ x$ (43)

with $h (x, p, t) k$ as our new unknown functions. They obey the simpler relations
$⌊ ⌋ ∫ ( )2 ∫ ∞ ∂ ρ ∂ ˜S 1 ∑ ∂ S˜ ∞ ∂ S˜ 3 ------- ⌈ ----- ------ ------- ⌉ 3 ----- - d x + + V = d x ρhk (x, , t). - ∞ ∂ xk ∂ t 2m ∂ xj - ∞ ∂ x j$ (44)

On a first look this condition for the allowed forces looks similar to the $p -$ independent case [see (36)]. But the dependence of $hk$ on $x, t$ cannot be considered as ’given’ (externally controlled), as in the $p -$ independent case, because it contains now the unknown $x, t$ -dependence of the derivatives of $S˜$ . We may nevertheless try to incorporate the r.h.s by adding a term $T˜$ to the bracket which depends on the derivatives of the multivalued quantity $S˜$ . This leads to the condition
$˜ ˜ ∂-S˜- ∫ ∞ ∂ S ∂ T (x, ∂x , t) 3 hk (x, ----- , t ) = - --------------------+sk , d x ρsk = 0. ∂ x ∂ x k - ∞$ (45)

But this relation cannot be fulfilled for nontrivial $h , T˜ k$ because the derivatives of $˜S$ cannot be subject to further constraints beyond those given by the differential equation; on top of that the derivatives with regard to $x$ on the r.h.s. create higher order derivatives of $˜ S$ which are not present at the l.h.s. of Eq. (45). The only possibility to fulfill this relation is for constant $˜ ∂S--- ∂x$ , a special case which has in fact already be taken into account by adding the mechanical potential $V$ . We conclude that the statistical constraint leads to $hk = T˜ = 0$ and that the statistical condition (44) takes the form
$⌊ ( ) ⌋ ∫ 2 ∂ ρ ∂ S˜ 1 ∑ ∂ ˜S - d3x -------⌈ ----- + ------ ------- + V ⌉ = 0. ∂ xk ∂ t 2m ∂ xj j$ (46)

Thus, only a mechanical potential and the four electrodynamic potentials are compatible with the statistical constraint and will consequently - assuming that the present statistical approach reflects a fundamental principle of nature - be realized in nature. As is well known all existing interactions follow (sometimes in a generalized form) the gauge coupling scheme derived above. The statistical conditions imply not only Schrödinger’s equation but also the form of the (gauge) coupling to external influences and the form of the corresponding local force, the Lorentz force.

In the present derivation the usual order of proceeding is just inverted. In the conventional deterministic treatment the form of the local forces (Lorentz force), as taken from experiment, is used as a starting point. The potentials are introduced afterwards, in the course of a transition to a different formal framework (Lagrange formalism). In the present approach the fundamental assumptions are the statistical conditions. Then, taking into account an existing mathematical freedom (multi-valuedness of a variable) leads to the introduction of potentials. From these, the shape of the macroscopic (Lorentz) force can be derived, using the validity of the statistical conditions as a constraint.

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