“Ontology” is a fancy philosopher’s word for “what really exists.” In general, for the pre-quantum sorts of theories we’ve been looking at, “what really exists” according to the theory is fairly obvious and non-controversial. For example, according to Newtonian mechanics, the world is made ofparticleswhich move under the influence of (gravitational and probably other) forces they exert on one another. The picture is similar according to Maxwellian electrodynamics: the world is made ofparticles which interact with one another by means of the electric and magneticfields. The ontology of Maxwellian electrodynamics, that is, includes particles and fields.
But there are often mathematically equivalent ways to formulate the basic laws of a theory, and this can sometimes raise questions about which formulation we should take seriously, as telling us in some relatively direct sense what really exists, physically, according to the theory.
For example, as you probably know, it is possible in electrodynamics to work with the so-calledpotentials(the scalar or electrostatic potential, and then the perhaps- less-familiar magnetic vector potential) instead of thefields. Let us briefly review some of this.
One of Maxwell’s equations – the so-called “Gauss’ Law for Magnetism” – reads
∇ · B =0. Since the divergence of any curl is identically zero, we can ensure that this equation is automatically satisfied if we introduce a magnetic vector potentialA related to the magnetic field by
B= ∇ × A. (1.29) This allows us to re-write Faraday’s Law as
∇ ×
E+∂A
∂t
=0. (1.30)
But then, because the curl of a gradient is identically zero, we can ensure that this equation is automatically satisfied if we introduce a “scalar potential”φsatisfying
E+∂A
∂t = − ∇φ. (1.31) (The minus sign is just for convention/convenience.) Equivalently, if we write the electric field in terms of the scalar and vector potentials as
E= − ∇φ−∂A
∂t (1.32)
we guarantee the satisfaction of Faraday’s Law. That then leaves the other two (non- homogeneous) Maxwell equations, which can now be re-written, in terms ofφand
Aas, respectively,
∇2φ+ ∂
∂t(∇ · A)= −ρ
0 (1.33)
and
∇2A− 1 c2
∂2A
∂t2 = −μ0j+ ∇(∇ · A+ 1 c2
∂φ
∂t). (1.34)
These both look a little messy and complicated, but actually we have not yet taken full advantage of the freedom afforded by the potentials. In particular, since all that is really required of the potentials is that their variousderivativesgive the fieldsE andB, there is an element of arbitrariness that we can leverage to make the equations look nicer.
For example, it’s clear from the defining relation B= ∇ × Athat (since, again, the curl of a gradient is identically zero!) we could change Aby the gradient of an arbitrary scalar function without affecting B. That is,
A→ A+ ∇λ (1.35)
leavesBunchanged.
“But” (I can hear you saying) “changing Ain this waywouldaffect the electric field!” That’s true, but we can “undo” that change byalsorequiring that, whenAis shifted as in Eq. (1.35), we also shift the scalar potentialφas follows:
20 1 Pre-Quantum Theories φ→φ−∂λ
∂t. (1.36)
It is easy to see that, if bothφandAare shifted in these ways, the fieldsBandEare unaffected. These “shifts” in the potentials are called “gauge transformations” and the idea is that there is a wholeequivalence classof potentials (corresponding to all possible scalar functionsλ) that correspond to the same field configurations.
This means that we are free to choosea particular set of potentials that makes some of the terms in Eqs. (1.33) and (1.34) disappear. We will briefly discuss two of these possible choices.
The first choice is to choose potentials satisfying the so-called “Lorentz gauge”
condition:
∇ · A+ 1 c2
∂φ
∂t =0. (1.37)
With this choice, the wave equations satsified by φand A take on the following particularly simple forms:
∇2φ− 1 c2
∂2φ
∂t2 = −ρ
0 (1.38)
and
∇2A− 1 c2
∂2A
∂t2 = −μ0j. (1.39)
These are nice because they are precisely of the general form, Eq. (1.22), that we investigated earlier. It’s clear, that is, that in Lorentz gauge, the effects of charges on the potentials propagate outward at the speed of light.
But here is another perfectly valid choice of gauges, the so-called “Coulomb gauge” in which
∇ · A=0. (1.40)
This turns out to imply the following dynamical equations (with source terms) forφ andA:
∇2A− 1 c2
∂2A
∂t2 = −μ0j+ 1 c2∇∂φ
∂t (1.41)
and
∇2φ= −ρ/0. (1.42)
Look in particular at the latter equation. It is just like Eq. (1.38), but with the prop- agation speedcset to infinity so that the term involving the second derivative with respect to time vanishes. So it implies that the scalar potentialφ(x,t)is determined by theinstantaneousconfiguration of chargesρ(x,t)– in just the same way that the gravitational force on a given particle in Newtonian mechanics is determined by the instantaneous configuration of other particles. So the objectφ– in Coulomb gauge
– is nonlocal! It changesinstantaneously, without any speed-of-light delay, if some distant charge is wiggled.5
But so what? We should only bebotheredby this kind of nonlocality, from the point of view of consistency with relativity, if we takeφas corresponding to something that is physically real. So does it? The usual answer is: no! The potentialsφand Aare mere calculation devices – it is instead the fieldsEand Bwhich we should take as directly corresponding to “physical stuff that really exists” according to the theory. To put this in terms of our notation from the previous section, it would be appropriate to letχstand for (say) the value ofEorBat some point, but it would not be appropriate to let χrepresent the value of φor Aat some point. In order to function as intended, it is important that theχs in Bell’s formulation of locality represent quantities that are endorsed, by the theory in question, as physically real.
Bell gives a memorable analogy here:
The situation is further complicated by the fact that therearethings whichdogo faster than light. British sovereignty is the classical example. When the Queen dies in London (long may it be delayed) the Prince of Wales, lecturing on modern architecture in Australia, becomes instantaneouslyKing. (Greenwich Mean Time rules here.) And there are things like that in physics. In Maxwell’s theory, the electric and magnetic fields in free space satisfy the wave equation
1 c2
∂2E
∂t2 − ∇2E=0, 1
c2
∂2B
∂t2 − ∇2B=0
...corresponding to propagation with velocityc. But the scalar potential, if one chooses to work in ‘Coulomb gauge’, satisfies Laplace’s equation
−∇2φ=0
...corresponding to propagation with infinite velocity. Because the potentials are only math- ematical conveniences, and arbitrary to a high degree, made definite only by the imposition of one convention or another, this infinitely fast propagation of the Coulomb-gauge scalar potential disturbs no one. Conventions can propagate as fast as may be convenient. But then we must distinguish in our theory between what is convention and what is not [6].
This is a point that will occupy us considerably in the coming weeks. In quantum theory, which objects in the formalism are we supposed to take seriously, as corre- sponding to things that are physically real, and which are (like the scalar potential and British sovereignty!) bound up in some way with human knowledge or conven- tions? In particular, what is the ontological status of the quantum mechanical wave function?
5For a nice analysis of how, in Coulomb gauge, the electric field retains its local character even though the scalar potential is nonlocal, see Ref. [7].
22 1 Pre-Quantum Theories