# Tag: classical physics

# Local Gauge Transformations in Locally Gauge Invariant Relativistic Field Theory

The question arises of whether local space-time symmetries – arbitrary co-ordinate transformations that leave the explicit form of the equations of motion unaffected – also have an active interpretation. As in the case of local gauge symmetry, * it has been argued in the literature that the introduction of a force is required to ‘restore’ local symmetry*. In the case of arbitrary co-ordinate transformations, the force invoked is gravity. Once again, we believe that the arguments (though seductive) are wrong, and that it is important to see why. Kosso’s discussion of arbitrary coordinate transformations is analogous to his argument with respect to local gauge transformations. He writes:

Observing this symmetry requires comparing experimental outcomes between two reference frames that are in variable relative motion, frames that are relatively accelerating or rotating….One can, in principle, observe that this sort of transformation has occurred. … just look out of the window and you can see if you are speeding up or turning with respect to some object that defines a coordinate system in the reference frame of the ground…Now do the experiments to see if the invariance is true. Do the same experiments in the original reference frame that is stationary on the ground, and again in the accelerating reference frame of the train, and see if the physics is the same. One can run the same experiments, with mechanical forces or with light and electromagnetic forces, and observe the results, so the invariance should be observable…But when the experiments are done, the invariance is not directly observed. Spurious forces appear in the accelerating system, objects move spontaneously, light bends, and so on. … The physics is different.

In other words, if we place ourselves at rest first in an inertial reference frame, and then in a non-inertial reference frame, our observations will be distinguishable. For example, in the non-inertial reference frame objects that are seemingly force-free will appear to accelerate, and so we will have to introduce extra, ‘spurious’, forces to account for this accelerated motion. The transformation described by Kosso is clearly not a symmetry transformation. Despite that, his claim appears to be that if we move to General Relativity, this transformation becomes a symmetry transformation. In order to assess this claim, let’s begin by considering Kosso’s experiment from the point of view of classical physics.

Suppose that we describe these observations using Newtonian physics and Maxwell’s equations. We would not be surprised that our descriptions differ depending on the choice of coordinate system: arbitrary coordinate transformations are not symmetries of the Newtonian and Maxwell equations of motion as usually expressed. Nevertheless, we are free to re-write Newtonian and Maxwellian physics in generally covariant form. But notice: the arbitrary coordinate transformations now apply not just to the Newtonian particles and the Maxwellian electromagnetic fields, but also to the metric, and this is necessary for general covariance.

Kosso’s example is given in terms of passive transformations – transformations of the coordinate systems in which we re-coordinatise the fields. In the Kosso experiment, however, we re-coordinatise the matter fields without re-coordinatising the metric field. This is not achieved by a mere coordinate transformation in generally covariant classical theory: a passive arbitrary coordinate transformation induces a re-coordinatisation of not only the matter fields but also the metric. The two states described by Kosso are not related by an arbitrary coordinate transformation in generally covariant classical theory. Further, such a coordinate transformation applied to only the matter and electromagnetic fields is not a symmetry of the equations of Newtonian and Maxwellian physics, regardless of whether those equations are written in generally covariant form.

Suppose that we use General Relativity to describe the above observations. Kosso suggests that in General Relativity the observations made in an inertial reference frame will indeed be related by a symmetry transformation to those made in a non-inertial reference frame. He writes:

The invariance can be restored by revising the physics, by adding a specific dynamical principle. This is why the local symmetry is a dynamical symmetry. We can add to the physics a claim about a specific force that restores the invariance. It is a force that exactly compensates for the local transform. In the case of the general theory of relativity the dynamical principle is the principle of equivalence, and the force is gravity. … With gravity included in the physics and with the windows of the train shuttered, there is no way to tell if the transformation, the acceleration, has taken place. That is, there is now no difference in the outcome of experiments between the transformed and untransformed systems. The force pulling objects to the back of the train could just as well be gravity. Thus the physics, all things including gravity considered, is invariant from one locally transformed frame to the next. The symmetry is restored.

This analysis mixes together the equivalence principle with the meaning of invariance under arbitrary coordinate transformations in a way which seems to us to be confused, with the consequence that the account of local symmetry in General Relativity is mistaken.

Einstein’s field equations are covariant under arbitrary smooth coordinate transformations. However, as with generally covariant Newtonian physics, these symmetry transformations are transformations of the matter fields (such as particles and electromagnetic radiation) combined with transformations of the metric. Kosso’s example, as we have already emphasised, re-coordinatises the matter fields without re-coordinatising the metric field. So, the two states described by Kosso are not related by an arbitrary coordinate transformation even in General Relativity. We can put the point vividly by locating ourselves at the origin of the coordinate system: I will always be able to tell whether the train, myself, and its other contents are all freely falling together, or whether there is a relative acceleration of the other contents relative to the train and me (in which case the other contents would appear to be flung around). This is completely independent of what coordinate system I use – my conclusion is the same regardless of whether I use a coordinate system at rest with respect to the train or one that is accelerating arbitrarily. (This coordinate independence is, of course, the symmetry that Kosso sought in the opening quotation above, but his analysis is mistaken.)

What, then, of the equivalence principle? The Kosso transformation leads to a physically and observationally distinct scenario, and the principle of equivalence is not relevant to the difference between those scenarios. What the principle of equivalence tells us is that the effect in the second scenario, where the contents of the train appear to accelerate to the back of the train, may be due to acceleration of the train in the absence of a gravitational field, or due to the presence of a gravitational field in which the contents of the train are in free fall but the train is not. Mere coordinate transformations cannot be used to bring real physical forces in and out of existence.

It is perhaps worthwhile briefly indicating the analogy between this case and the gauge case. Active arbitrary coordinate transformations in General Relativity involve transformations of both the matter fields and the metric, and they are symmetry transformations having no observable consequences. Coordinate transformations applied to the matter fields alone are no more symmetry transformations in General Relativity than they are in Newtonian physics (whether written in generally covariant form or not). Such transformations do have observational consequences. Analogously, local gauge transformations in locally gauge invariant relativistic field theory are transformations of both the particle fields and the gauge fields, and they are symmetry transformations having no observable consequences. Local phase transformations alone (i.e. local gauge transformations of the matter fields alone) are no more symmetries of this theory than they are of the globally phase invariant theory of free particles. Neither an arbitrary coordinate transformation in General Relativity, nor a local gauge transformation in locally gauge invariant relativistic field theory, can bring forces in and out of existence: no generation of gravitational effects, and no changes to the interference pattern.

# Matter Defined as Just Another Quantum State: Whatever Ontologies.

In quantum physics, vacuum is defined as the ground state of a quantum field. It is a state of minimum energy, corresponding to zero particles. Note that this definition of vacuum uses already the conceptual and formal machinery of quantum field theory. It is justifiable to ask weather it is possible to give a more theory-independent definition with lesser theoretical load. In this situation vacuum would be an entity which is explained – not just defined within and then explored – by quantum field theory. For example, one could attempt an operational definition of vacuum as the state in which no particles are detected. But then we have to specify how to detect the particles, with what efficiency, etc., that is, we need a model for the particle detector. Such a model, known as the Unruh-DeWitt detector, is constructed however from within quantum field theory. Unruh-DeWitt detector is a simplified model of a real particle detector. Its basic property is the fact that it is linearly coupled to the field, so that it can detect one-particle states. Indeed, as long as the detector moves inertially in Minkowski spacetime, it really does react to one-particle states and not to the 0-particle state (vacuum). However, when it moves non-inertially, it may react even in the vacuum. The energy needed for the reaction in the vacuum comes from the agency that accelerates the detector (not from the vacuum energy).

The vacuum is simply a special state of the quantum field – implying that quantum physics allows the return of the concept of ether, although in a rather weaker, modified form. This new ether – the quantum vacuum – does not contradict the special theory of relativity because the vacuum of the known fields are constructed to be Lorentz-invariant. In some sense, each particle in motion carries with it its own ether, thus Lorentz transformations act in the same way on the vacuum and on the particle itself. Otherwise, the vacuum state is not that different from any other wavefunction in the Hilbert space. Attaching probability amplitudes to the ground state is allowed to the same degree as attaching probability amplitudes to any other state with nonzero number of particles. In particular, one expects to be able to generate a real property – a value for an observable – in the same way as for any other state: by perturbation, evolution, and measurement. The picture that quantum field theory provides is that both particles and vacuum are now constructed from the same “substance”, namely the quantum states of the fields at each point (or, equivalently, that of the modes). What we used to call matter is just another quantum state, and so is the absence of matter – there is no underlying substance that makes up particles as opposed to the absence of this substance when particles are not present. One could even turn around the tables and say that everything is made of vacuum – indeed, the vacuum is just one special combination of states of the quantum field, and so are the particles. In this way, the difference between the two worldviews, the one where everything is a plenum and vacuum does not exist, and the other where the world is empty space (nonbeing) filled with entities that truly have the attribute of being, is completely dissolved. Quantum physics essentially tells us that there is a third option, in which these two pictures of the world are just two complementary aspects. In quantum physics the objects inhabit at the same time the world of the continuum and that of the discrete.

Incidentally, the discussion has implications for the concept of individuality, a pivotal one both in philosophy and in statistical physics. Two objects are distinguishable if there is at least one property which can be used to make the difference between them. In the classical world, finding this property is not difficult, because any two objects have a large amount of properties that can be analyzed to find a different one. But, because in quantum field theory objects are only combinations of modes, with no additional properties, it means that one can have objects which cannot be distinguished one from each other even in principle. For example, two electrons are perfectly identical. To use a well-known Aristotelian distinction, they have no accidental properties, they are truly made of the same essence.

To see in a simple way why quantum physics requires a re-evaluation of the concept of emptiness, the following qualitative argument is useful: the Heisenberg uncertainty principle shows that, if a state has a well-defined number of particles (zero) the phase of the corresponding field cannot be well-defined. Thus, quantum fluctuations of the phase appear as an immediate consequence of the very definition of emptiness. Another argument can be put forward: the classical concept of emptiness assumes the separability of space in distinct volumes. Indeed, to be able to say that nothing exists in a region of space, we implicitly assume that it is possible to delimitate that region of space from the rest of the world. We do this by surrounding it with walls of some sort. In particular, the thickness of the walls is irrelevant in the classical picture, and, as long as the particles do not have enough energy to penetrate the wall, all that matters is the volume cut out from space. Yet, quantum physics teaches us that, due to the phenomenon of tunneling, this is only possible to some extent – there is, in reality, a non-zero probability for a particle to go through the walls even if classically they are prohibited to do so because they do not have enough energy. This already suggests that, even if we start with zero particles in that region, there is no guarantee that the number of particles is conserved if e.g. we change the shape of the enclosure by moving the walls. This is precisely what happens in the case of the dynamical Casimir effect. These demonstrate that in quantum field theory the vacuum state is not just an inert background in which fields propagate, but a dynamic entity containing the seeds of multiple possibilities, which are actualized once the vacuum is disturbed in specific ways.