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Next: 11.4 Quantum Substances Up: 11. Two Stages of Previous: 11.2 Actual and Virtual



11.3 `Gauge' Invariances

It is only with virtual events that interactions occur, as actual events are purely selections of a wave function that already includes all possible interactions. If we consider interactions as they are presumed to occur by quantum theory, then it is notable that there are many ways of describing their possibilities, ways which turn out to be describing essentially the same interactions, as they result in the same propensities of actualising, and hence in exactly the same probabilities for actual events. There are several features of interactions which behave in this way:
      Interactions imply extensive relations in space and time, rather than definite and absolute positions.       This is the old problem of Leibniz's `relational' view of space and time, versus Newton's `absolutist' view. It makes no difference to any interaction or event if the whole universe were to be `shifted over' in space, or `delayed' arbitrarily in time, or even (see Barbour [1982]) `rotated' about any axis.
    Interactions use potentials which are arbitrary with respect to additions of certain kinds of spatio-temporal functions, and hence generate complex-valued wave functions which have a corresponding arbitrariness in their phase.
    Interactions in quantum field theory operate in arbitrarily small regions of space-time, and hence almost always need a variable renormalisation to correct the self-energies (i.e. masses) of the particles, and to correct the coupling-strengths of the interaction terms.
    Interactions are sometimes between identical particles, for which, as they can be arbitrarily interchanged, we need to apply the Pauli Exclusion Principle.
These features, to be described in more detail below, mean that the many-body wave function $\Psi ( \vec{x}_1 , \vec{x}_2 , \ldots ,t)$, while seeming a perfectly ordinary and well-defined function in elementary quantum physics, is in fact defined from a domain of equivalent configuration spaces to a range of equivalent combinations of electromagnetic potentials and complex phases. Furthermore, it satisfies an equation (or Lagrangian) with masses, coupling constants and normalisations that are all only defined up to an equivalence class. It may appear that because of all these features that it is hardly a function that describes any real physical property at all, but that conclusion would be too sweeping. What we are seeing is how certain physical potentialities are defined over spaces of possibilities, and how the resulting propensities are described. We can always apply our process logic, even if we end up with unconventional propensities defined over unconventional spaces. We must not force the world to be simpler than it really is.


1. Equivalent Configuration Spaces

We discussed in section 8.2 how any acting of one substance on another, or any interacting, presupposes that the two substances stand in some kind of relation to each other. It does not matter what their absolute positions may be, but it is an essential prerequisite for the possibility of any interaction that there be what Whitehead and Leclerc call `extensive relations' between various substances.     Furthermore, this must in some sense be a real relation, if the substances are really to interact: they cannot be the merely phenomenal or ideal relations as Leibniz wanted to see them. Leibniz of course was the advocate of an overall `relational' view of space and time, versus Newton's `absolutist' view of space having definite and individual positions. In order to construct a theory of relational spacetime, extensiveness was thus defined in section 8.2 as a fundamental real relation between places. We thought of it as specifying absolutely the metric distance between any two points in space-time.

Despite our relational view of space and time, the many-body wave function $\Psi ( \vec{x}_1 , \vec{x}_2 , \ldots ,t)$ was still defined over the configuration space $( \vec{x}_1 , \vec{x}_2 , \ldots ,t) $as if the coordinates $\vec{x}_p$ defined absolute positions in an absolute space. It is certainly more convenient in practice to work with coordinates rather than relative distances in defining places, but if we insist on doing so, we must be careful to define our propensity fields only on certain equivalence classes of configuration spaces.   Barbour [1982, 1986] describes how this might be done.

It makes no difference to any interaction or event if the whole universe were to be shifted over in space, rotated about any axis, or delayed arbitrarily in time. According to the relational view of space and time, these cannot be physical changes: if we have only relational distances, they cannot even be described. If, however, we insist on using the coordinates $\vec{x}_p$ and t, then they do make a difference. Given all pairs of relational distances, for example, those coordinates are determined (in the non-relativistic case) only up to a seven-parameter `gauge transformation'

$\displaystyle \vec{x}_p ~$ $\textstyle \rightarrow$ $\displaystyle ~~ R \vec{x}_p + \vec{g}$ (11.4)
t    $\textstyle \rightarrow$    t + h (11.5)

where R is an orthogonal 3 $\times$ 3 matrix, g is a 3-component vector, and h is a scalar. The matrix R describes rotations of the coordinate system, the vector g describes translations of its origin, and h sets the origin of the time coordinate. Furthermore, different rotations and translations can be used at different times:
$\displaystyle \vec{x}_p (t) ~~$ $\textstyle \rightarrow ~~ R(t) \vec{x}_p (t) + \vec{g} (t) ,$   (11.6)

which allows the universe as a whole to be described by a coordinate system in which it (apparently) has an angular momentum and/or a linear velocity. Barbour also considers the case where the `time shift' h varies with time: this is equivalent to replacing the time coordinate t by any other coordinate $ t ^\prime ,$ subject only to the requirement of monotonic increases: $ d t ^\prime / dt > 0.$  Similar laws of transformation hold for relativistic space-time, based on the inhomogeneous Lorentz group.

These transformations are all those allowed which do not change a given physical situation. The set of all ordered pairs $( \vec{x}_1 , \vec{x}_2 , \ldots ,t) $ which can be transformed among themselves by equations (11.4) and (11.6) therefore form an equivalence class. It is the specification of which equivalence class which describes the physical situation, and all physical laws should depend only on the equivalence class, not on any preferred members of the class. That is, it should not make any difference to physical processes or physical laws if the universe as a whole were described by new coordinates which were translated and/or rotated compared with any other coordinate system. Absolute positions and rotations have no significance. We say then, that the physical laws must be invariant under the transformations (11.4) and (11.6).

The relational view of space-time can therefore be expressed either as extensiveness being purely a relation between places, and places not being individuated beyond their extensive relations, or as space-time being labelled by configuration coordinates $ ( \vec{x}_p , t), $ with there being additional laws of `gauge invariance' which state that all physical laws and probabilities should be invariant under the transformations of equations (11.4) and (11.6).  


2. Equivalent Phases and Potential Gauges

The movement of charged particles in an electromagnetic field is determined by the electric field vector E and magnetic field vector B.   Both of these, because they satisfy Maxwell's field equations, can be derived from a four-component `vector potential' (V,A) by
$\displaystyle \vec{B}$ = $\displaystyle \nabla \times \vec{A}$ (11.7)
$\displaystyle \vec{E}$ = $\displaystyle - \nabla V - \frac{\partial \vec{A}}{\partial t} .$ (11.8)

However, as explained in any textbook on electromagnetic theory, the potentials A and V are not uniquely determined by the equations (11.7). For B is left the same if A changes by the gradient of some function:  
$\displaystyle \vec{A} ~~ \rightarrow ~~ \vec{A} ^\prime = \vec{A} + \nabla \chi$     (11.9)

while E is then also unchanged if V changes by
$\displaystyle V ~~ \rightarrow ~~ V ^\prime = V - \frac{\partial \chi}{\partial t} .$     (11.10)

Since, in the Maxwell theory, the physically measurable quantities are E and B, and these do not change under the transformations (11.9) and (11.10), we have an invariance of the theory. The transformations (11.9) and (11.10) are called gauge transformations, and are an essential feature of Maxwellian electromagnetic theory. Furthermore, the function $\chi$ may be an arbitrary function of space and time: $\chi( \vec{x} ,t). $ The time-derivative of $\chi$, for example, acts like the origin of potential scale V.

  It is well known in physics that only differences of potentials have any physical significance, and that absolute values of potential fields are not strictly meaningful. This fact can be expressed more formally by the requirement that all the laws of physics, and all event probabilities, are invariant under the gauge transformations (11.9) and (11.10). Not only do we have a global invariance of the physical laws under changes in the meaning of V = 0 in electrostatics, but the gauge transformations express a local invariance of the laws under potential changes that may differ at every point in space and time. Local invariance imposes a much more stringent condition on physical laws. In the previous section, equation (11.4) expresses a global invariance, whereas equation (11.6) expresses an invariance that is local at least in time (though not in space), and hence has more physical content.

We now look at how gauge invariance is incorporated into quantum mechanics.   For a particle of mass m and charge q, the familiar non-relativistic Schrödinger's equation (11.1) becomes

$\displaystyle - \frac{\hbar^2}{2m} (\nabla - i q \vec{A} / \hbar)^{2} \psi + q V \psi
= i \hbar \frac{\partial \psi }{\partial t}.$     (11.11)

If Maxwell's theory and quantum mechanics are not to be in conflict, there must not be any observable effects on the wave function $\psi$ of the gauge transformations (11.9) and (11.10). The equation (11.11) remains the same if, when the gauge transformations are performed, the wave function is also transformed according to
$\displaystyle \psi ~~ \rightarrow ~~ \psi ^\prime = e ^ {i q \chi ( \vec{x} ,t)/ \hbar} \psi .$     (11.12)

where we see explicitly that the local nature of the gauge transformations is to change the phase of the wave function by a correspondingly suitable amount. The transformation (11.12) is also perfectly valid for relativistic wave functions. The conclusion is that the local choice of the phase of the wave function is linked to the choice of the vector potential (V,A).

One could at this point take a positivistic attitude, and say that the potentials V and A. were `theoretical constructs', as the empirically measurable quantities are the field vectors E and B.   This argument would be reinforced by the unobservability of phases in quantum mechanics: all probabilities depend on the square moduli $ \mid \psi \mid ^{2} ,$ with only relative phases having any observable consequence. The response to this argument is to note that relative phase are observable, just as are relative potentials, but that does not make them any the less real. If we attribute reality to the equivalence classes of potential/phase combinations, with the equivalence classes being defined by the possibility of gauge transformations between all members of each class, then potentials and phases have this new kind of `relational reality'. The situation is exactly analogous to the spatio-temporal invariances of the previous section. Although the absolute position, time, velocity and angular momentum of the universe are physically unobservable, relative positions, times, velocities and angular momenta are still perfectly real and definable.

The point at issue is (interestingly) not merely philosophical. For sometimes the field potentials B and E can be vanishingly small even though the potentials V and A may be significant. Because it is V and A which enter into the Schrö.dinger's equation (11.11), the propensities of particles can still be affected in those situations.   Such is the case with the Aharanov-Bohm effect     (Aharanov & Bohm [1959]), where an electron affected by the relative changes in the vector potentials on either side of a long thin solenoid. There is hence a phase change between the two paths, and this leads to observable interference patterns. Furthermore, these effects persist even when the fields E and B become vanishingly small, so if reality was only to be granted to these fields, then there would be an unusually disproportionate ratio between causes and effects in this case   (see Berry [1980, 1986] for more discussion here). Taking V and A as causally efficacious produces a better `balance' between the magnitudes of causes and of effects.

3. Equivalent `Infinitesimal' Sizes of Virtual Events

Figure: Some of the virtual events which contribute to an electron's self-energy.

      In quantum field theory, virtual processes such as those shown in 11.3 contribute to the effective mass and charge of the particles involved. If the theory is naively extended to include virtual events of smaller and smaller extensions, down to the limit of point events, then there is the embarrassing consequence of infinite contributions to the apparent masses and charges. Physicists have come up with a process called `renormalisation', which involves essentially putting in corrections to these effects so that the final masses and charges come out to be what we observe them to be. The trouble is that, strictly speaking, these corrections have to be infinite!   No-one has been very happy with this method -- Feynman [1985] says that `no matter how clever the word, it is what I would call a dippy process!'. However, physicists have grown accustomed to the idea, and even make it a necessary feature of the new theories they are constructing to describe the weak and strong nuclear interactions.     The Weinberg-Salaam and quantum chromodynamics (quark) theories are attractive to them precisely because they do allow a renormalisation process to be performed, and finite results obtained.

All the infinite changes come essentially because virtual events are considered to have proximities and sizes of all values, down to and including zero distances. Physicists feel that `one should be able to go down to zero distance in order to be mathematically consistent' (Feynman [1985]), but that is where the trouble is. Then,

instead of including all possible coupling points [i.e. virtual events] down to a distance of zero, if one stops the calculation when the distance between coupling points is very small -- say, 10-30 centimeters, [very much] smaller than anything observable in experiment (presently 10-16 centimeters) -- then there are definite values for n and j [the initial mass and charge numbers] that we can use so tha the calculated mass comes out to match the mass m observed in experiments, and the calculated charge matches the observed charge, e Now, here's the catch: if somebody else comes along and stops their calculation at a different distance -- say, 10-40 centimeters -- their values for n and j to get the same m and e come out different!

$\ldots$ later, Bethe and Weisskopf noticed something: if two people who stopped at different distances to determine n and j from the same m and e then calculated the answer to some other problem -- each using the appropriate but different values for n and j -- their answers to this other problem came out nearly the same! In fact, the closer to zero distance the calculations for n and j were stopped, the better the final answers for the other problem would agree!

So it appears that the only things that depend on small distances between coupling points are the values for n and j -- theoretical numbers that are not directly observable anyway; everything else, which can be observed, seems not to be affected. People could finally calculate with the theory of quantum electrodynamics!11.1


I do not wish to condone an ad hoc process which may eventually be done away with   (e.g. if `superstring' theories11.2 become tractable), but it does seem to me that a fundamental misunderstanding is obstructing the way to a possible realistic interpretation of what quantum field theory could be about.     The problem goes back to the question of chapter 6 of whether space (and time) are actually composed of points of zero size (albeit in a transfinite number), or whether the continuity and point structure of space and time should be seen as a process of ever more possibilities for subdivision, without there being a definite end. Perhaps there is a difference between a continuum of a bounded (though transfinite) number of points, and a continuum of a unbounded possibilities for division.

  If we adopt the latter view, virtual events (or `coupling points') should not be seen as necessarily occurring at all separations down to zero, but they should be seen as occurring in regions which are arbitrarily small, without there being a minimum size. We do not want a minimum size, whether it is zero (as Feynman wants), or non-zero (as if space were made of finite `lumps'). We therefore want physical laws to be independent of this precise arbitrariness, and so can allow a physical theory to be a particular equivalence class of combinations of arbitrarily-small virtual events with correspondingly suitable initial coupling constants (such as the `bare' mass n and `bare' charge j). It would be interesting to speculate at this point as to whether there would be any new physical content in the requirement of `local renormalisability'. This is the requirement that all virtual events are independent in their arbitrariness of size, and that physical laws should be unaffected by all these variabilities.

In this way, the door is opened for a possible realistic understanding of the renormalisation process, should it prove to be a permanent feature of physical theories. I am not claiming that it is necessarily part of physics, as our philosophy of nature is only providing a general framework in which quantum-like theories can be formulated realistically, and is not deriving them specifically.    


4. Equivalent Exchanges of Identical Particles

Interactions are sometimes between identical particles, for which we need to apply the Pauli Exclusion Principle. This states that two identical particles cannot be in the same propensity state, and limits the allowable many-body wave functions $\Psi ( \vec{x}_1 , \vec{x}_2 ,
ellipsis ,t).$ I do not know why the Pauli Principle should hold, but if it is true, then it can be also framed within our theory of equivalence classes.   Following Leinas et al. [1977], the configuration space $( \vec{x}_1 ,
\vec{x}_2 , \ldots \vec{x}_N ,t) $ of an N-particle system is set up while identifying sets of configurations which can be transformed into each other by the exchange of identical particles. That is, configuration space consists of equivalence classes of physically distinguishable configurations. Within each equivalence class, as physically indistinguishable, are those configurations that are `really the same' because identical particles cannot be distinguished.

Leinas et al. show that if space has three or more dimensions, and if $\Psi ( \vec{x}_1 , \vec{x}_2 , \ldots ,t).$ is an analytic function of its variables, then the wave function $\Psi$ must be either symmetric  

$\displaystyle \Psi( \ldots , \vec{x}_p , \ldots , \vec{x}_q , \ldots ,t) =
+ \Psi( \ldots , \vec{x}_q , \ldots , \vec{x}_p , \ldots ,t)$     (11.13)
$\displaystyle \mbox{ for all } 1 ~ \leq ~ p, q ~ \leq N \mbox{ and } p,~q \mbox{ bosons, }$     (11.14)

    or anti-symmetric
$\displaystyle \Psi( \ldots , \vec{x}_p , \ldots , \vec{x}_q , \ldots ,t) =
- \Psi( \ldots , \vec{x}_q , \ldots , \vec{x}_p , \ldots ,t)$     (11.15)
$\displaystyle \mbox{ for all } 1 ~ \leq ~ p, q ~ \leq N \mbox{ and } p,~q \mbox{ fermions, }$     (11.16)

nder the process of exchanging identical particles. This is because, in three or more dimensions, the trajectory of the process of exchanging particles any even number of times can be continuously deformed and contracted into the `identity' process of zero exchanges. The trajectory for any odd number of exchanges can be similarly deformed into a trajectory for just one exchange. This means that the $\Psi$, as a multi-valued analytic function, falls into one of the categories (11.13) or (11.15) above. These are the categories of bosons and fermions respectively. For fermions, the Pauli Exclusion Principle follows from equation (11.15). One can see immediately, for example, that the wave function must be zero if two fermions are in the same place:

\begin{displaymath}{ \Psi( \ldots , \vec{x}_p , \ldots , \vec{x}_q , \ldots ,t) ...
... 1 ~ \leq ~ p, q ~ \leq N \mbox{ and } p,q \mbox{ fermions. }}

Leinas emphasises that within this formulation, the two possibilities (of bosons or fermions) are singled out in a natural way, and not as the consequence of any symmetrisation postulate.  


What we have done in this section is to point out that the space of `virtual events', and the fields of propensities that thereby result, are not definite functions over Newtonian configuration spaces to complex-valued field functions for definite masses and charges. If we do want to keep these features of our description, all these `spaces' and field descriptions should be seen as simply particular (arbitrary) members of certain equivalence classes. It is the specification of particular equivalence classes which really describes the spaces and propensities for virtual events. From modern physics, it appears to be as essential feature of the `space' of possibilities for virtual events that it is defined in this way.

I am not claiming to have found reasons why nature should be like this, only to have put the four features above (usually regarded as some of the more mysterious parts of physics) on a uniform footing. When actual and virtual events are distinguished in a two-stage process, with virtual events being the production of propensities for actual events, we can allow that the two kinds of events occur in different `spaces': different sets of possibilities. With the help of the arguments in the previous chapters, we can begin to see the manner in which events, possibilities and propensities etc. can be said to exist. The philosophy of nature is useful as it shows how these things can exist, and thus provides an ontological framework in which quantum physics (and quantum field theory) can be interpreted realistically.  

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Next: 11.4 Quantum Substances Up: 11. Two Stages of Previous: 11.2 Actual and Virtual
Prof Ian Thompson


Author: I.J. Thompson (except as stated)