Relative QM measurment and the ontology of space-time structure

0:00 During the interval from T1 to T2 measurement, M1 takes place. During the interval from T3 to T4 measurement, M2 takes place. If the original ensemble of systems is in the pure state psi, then after acquiring the ith result in M1, the ith sub-ensemble is represented by a pure state psi sub i, and then that is subject to the M2 measurement, which yields some result j and the final state vector for the system. Now, in the non-relativistic domain, one can very easily answer the question of what are the states that represent this system. Before I answer that question, which actually pertains to the lower diagram, let me make a reference to the top diagram. There is a common theory of interest that is confined to a bounded region of space-time, and I'm representing such bounded regions of space-time by these red circles. And in the upper diagram I want to indicate the possibility that the region of space-time in which the coupling to the apparatus takes place may be such that we could actually have what we would want to call two measurements taking place, And finally, we have the T1 sub-interval, during which we could say that the first measurement had finished and left us with a pure state, which is then subject to the second measurement. There would be some coupling to apparatus going on at all times here, and we would not be able to say anything about the state of the various sub-ensembles until both of those interactions are completed. During the coupling of the system of interest to the measuring apparatus, we cannot, in general, assign any pure state whatsoever to the system of interest. Now, in the lower diagram, I've indicated how the process looks from the point of view of a relatively moving Galilean inertial frame of reference. The fact that the green time axis of the relatively moving Galilean frame is tilted is the indication that that frame is indeed moving,

2:30 The fact that the green spatial axis is in the same direction as the black spatial axis is the indication that simultaneity of events is assessed in exactly the same way in the two Galilean frames with the same conclusions being reached by either Galilean observer. And in every instance, we find that whenever there is a situation that can be represented by a pure state in the original frame of reference, there is a corresponding pure state representation in the green frame of reference obtained by a unitary transformation from the initial frame. Psi is converted into Psi prime by U. Psi sub i, the state occurring after the first measurement takes place, is converted into Psi sub i prime by the same U because the relationship The two Galilean frames remains the same when psi sub ij is converted into psi sub ij prime by the same u. In a relativistic domain, one immediately encounters the first indication of this paradox was for a block on my list. I can't find it. It's somewhere in the 60s and 70s. I think around 1960. The block paradox occurs in the following way. There are three frames of reference now related to one another by Lorentz. The blue frame of reference moving off to the right, off in the direction of positive x relative to the black frame of reference. The green frame of reference moving off in the direction of negative x relative to the black frame of reference. And I'm presuming the existence of some measurement producing interaction confined to the space-time region represented by R. Let's for simplicity assume that the physical system that is being subject to this measurement interaction is a single particle system, and that therefore in general, certainly in the absence of measurement interactions, we would expect to be able to describe that single particle system by position-dependent wave functions, state functions, and if I want to be precise about that, I could talk about Newton-Wigner representation state functions, position representation state functions. Which have a direct probability interpretation, albeit they have some bizarre properties.

5:00 Now, let me suppose that I consider a point in space-time somewhere about there. And I ask questions about the state function of this single particle evaluated at that particular point. The function of space-time occurs at a time that is simultaneous with part of the points of space-time occupied by the measurement interaction region. As a consequence of that, in general I cannot expect there to be any pure state function associated with the properties of the particle at that point in space-time because the whole system is coupled with the measurement interaction. So in general I would have to expect there might be peculiar circumstances in which this is the case, but in the vast majority of circumstances one can imagine it would be the case. But in fact, the state function psi of x0 and t0, which is supposed to be the spacetime coordinates at that point in the black frame of reference, it just simply does not exist. There is no substitute. On the other hand, from the point of view of the blue frame of reference, the point t0 in spacetime occurs earlier than any of the points during which the measurement interaction takes place, because simultaneity is assessed by the blue frame of reference along lines that are parallel to the spatial axis of the blue frame of reference. And P0 is clearly therefore earlier than R and associated with a time coordinate zero prime. In that frame of reference I do expect that there will be a state function for the single particle and I can evaluate its amplitude at the point x zero prime t zero prime. However that state function evaluation cannot depend on the result of the measurement because in that frame of reference On the other hand, again, from the standpoint of the green frame of reference, the point P0 occurs later than the measurement interaction, the measurement interaction has been completed, the time coordinate of P0 is T0 double-point, because these are the lines along which the green observer, or the green frame of reference, assesses simultaneity. So in the green frame of reference, once again, there will be a state function for the particle associated with that point, and I can evaluate that, but whatever the result is, it will in general depend upon the result in the measurement. Now, my claim is that in a case like that, in which the black frame psi doesn't exist, in the blue frame it exists but does not depend on the measurement, in the green frame it exists but does depend upon the measurement,

7:30 Let's make this kind of paradox appear more realistic by considering the old familiar holistic aspects of the EPR paradox. Here I have an optical bench and at either end of the optical bench I have linear polarizers which I have deliberately indicated are oriented at some relatively elite I will call the direction of orientation here plus the direction of orientation of that one plus prime. I have something like say a sample of k and send photons That initial photon entangled state can be written, because of its peculiar spinless nature, it can be written in any one of an infinite number of different ways, and I have decided here to write it in those particular ways which make special reference to either the polarizer at the right end of the optical bench or the polarizer at the left end of the optical bench. If in particular the right-hand photon goes through, that means that this state is selected for the right-hand photon, and since it's necessarily coupled with UL-, this would be the state of the left photon if it didn't have to go through any polarizer. And if the right-hand photon doesn't pass through that polarizer, this state would have been selected. That photon would have been in that state if it were. This is the analysis that plays the same relationship to that, but this analysis of the... Any one of four possible things can happen. Both photons can get through the polarizers because of their relatively oblique orientation, or the right photon can get through and the left photon cannot, or the left photon and the right photon not, or neither photon. The arguments that I'm going to go through can be changed in the appropriate way for either of those four possible alternatives. I want to focus on the case in which both photons pass through the polarizers.

10:00 If both photons pass through the polarizers, then the final state of the two-photon system is UR plus UL prime plus. The right-hand photon goes through, meaning that it has that polarization plus. The left-hand photon goes through, meaning that it has that polarization plus minus. If the angle between the two polarizers was zero, of course, this could emphasize that we have a transition as a consequence of this measurement. From the singlet state prior to either photon encountering the polarizers to this product state. Tangled state to the product state after the photons have encountered the polarizers. One step closer. Now they suddenly realize. And we consider now the standpoint of the observer on the ground. From the standpoint of the observer on the ground, that observer would also agree that the emitted photons are in a singlet state, And finally, the black bar frame of reference to the ground frame of reference of the original singlet states are written in the same way. And it can be written in either terms of single photon states associated with that polarizer or with the third. However, now, it is clearly the case that the left moving photon encounters its polarizer prior to the right moving photon. And therefore, if I assume that both photons are ultimately going to go through the polarizers, it follows that But first, I have the left photon going through its polarizer, which immediately picks out this state. The left photon has gone through, so the right photon, which has not yet encountered its polarizer, is forced into the U-R prime minus state. For a short period of time, until the right photon encounters its polarizer, whereupon the left photon, being untouched by that polarizer, is left in the state, single-particle state that it was already forced in, and the right photon goes into that state because of our hypothesis that both photons can go through. Again, I remind you that if we considered any of the three other possibilities, we could go through similar kinds of arguments. They would be different in detail, but there would be corresponding stages in them.

12:30 In the second frame of reference, there is a time interval during which the r-photon is certain to be in the minus prime state, and no such time interval exists in the flat-par frame. Therefore, it appears that in the flat-par frame, there is no state, which is the unitary transform of that one. That's an apparent paradox. Let's look at this apparent paradox in a space-time diagram, which, alas, even though it is in one... We have initially this entangled state which can be written in either of these two ways for the two photons. These are the two photons written distressingly classically as though they had well-defined trajectories, but approaching their . . . In the green frame of reference, which is in fact the ground frame, there is this time interval during which there is an intermediate product state as a consequence of the left moving photon having encountered its polarizer prior to the right moving photon. So in the black frame, the entangled state goes suddenly into that state. In the green frame, the Lorentz transform of the entangled state goes temporarily into that state and then into the Lorentz transform of that state. If we looked at a blue frame, which was moving in the opposite direction relative to the ground, there would be an intermediate period during which the two photon system would move.

15:00 I myself, independently, and Giovannini, a substantivist of space-time, what my ultimate punchline will be, I think, should make you change your conception of the structural properties of that geometrical conceptual atom in which you believe in the substantial nature of. If you're a relationist, then my ultimate punchline should make you, if you believe me, if you accept my ultimate punchline, Should make you change your conception of the locus of relationships that that geometrical atom possesses. One must assign state vectors in the Heisenberg picture to all of the possible space-like hyperplanes. From the standpoint of any frame of reference, in any single frame of reference, one must make assignments of state vectors to all the possible space-like hyperplanes. But there are no measurements going on. You make the same assignment of state vectors to all hyperplanes. The state vector in the Heisenberg picture represents the entire evolutionary history of the quantum mechanical system. In the ordinary way of looking at things, it does not change with time. All the time dependence is carried by the observables and their eigenvectors. And I would simply be saying here a trivial thing, that time independence now becomes hyperplane independence for the state vector. Nothing has really changed, although the way in which you would describe the time dependence of the dynamical variables would change. I contend, and will be discussed at a later time, that in the absence of measurements this is saying absolutely nothing new whatsoever about relativistic quantum mechanics.

17:30 It is simply emphasizing a structural feature which is already in the theory and which is usually ignored in practice, but which our attention was already brought to by very early work by Schwinger and Tominaga, who in fact wanted to make these assertions for arbitrary curvilinear hypersurfaces, space-like hypersurfaces. I can find the conjecture to hyperplanes because that is quite sufficient to guarantee relativistic invariance and does not presuppose a commitment to local quantum field theory, which Schwinger and Tomanova were especially interested in. In the presence of measurements, things change a bit. In the presence of measurements, to all those hyperplanes that are prior to a measurement interaction region, in the Heisenberg picture, you assign... You assign the same state vector to all those hyperplanes that are prior, if as a result of the measurement you get the ith result, then to all those hyperplanes that are later than the measurement region, in a sense which I think is obvious. You assign the same state vector, and it is the so-called ith state vector, the eigenvector of the measured observable. These black, green, and blue lines in the previous diagrams, these black, green, and blue lines are not lines referring to different frames of reference. They are lines referring to a geometrical construct in Minkowski spacetime that can be used, and I am claiming should consistently be used, by observers in any given frame of reference. In what we previously called the black frame of reference, these hyperplanes, the black ones, would appear instantaneous, they would have definite values of the time coordinate associated with them, the green and blue ones would not, but there would be other frames of reference in which the blue ones appeared instantaneous, others yet in which the green ones appeared instantaneous. My point is that that's irrelevant for the use of this. The whole point is that you're referring to invariant geometrical constructs by which you make the assignment of state vectors. In particular, if you now look at this assignment, you see that there is always a possibility in a situation like this of making the unitary transform from the black frame, if the black state vector is now referred to the state vectors you would use on these hyperplanes, in the black frame of reference, which is now represented by a dashed line,

20:00 Then the transformation that you would make if you wanted to find the state vector that the green frame of reference would assign to the green hyperplane, to the hyperplanes that are prior to the matrix interaction region, you would make the green unitary transform, and you would make it here as you would make it up here for the state. And if you want to find the state vectors that would be assigned to all the hyperplanes in the blue frame of reference, the frame of reference in which the blue hyperplanes happen, In any frame of reference, the same state and the same state is assigned. This has the peculiar feature that it forces you, since different orientations always intersect in some points. In this 1 plus 1 spacetime diagram that I'm using, they all intersect in only one point. In 3 plus 1 dimensional spacetimes, differently oriented space-like hyperplanes would intersect in moving lines. But there would always be certain points of Minkowski spacetime common to differently oriented, any pair of differently oriented space, like hyperplanes. And this means, among other things, that if we go back to the kind of consideration that I discussed in connection with the block paradox, and focus on the state of the quantum mechanical system at a given spacetime point, The state function that we must associate with the system at that point depends crucially upon the hyperplane that is passing through that point that we wish to consider, and the specification of the state function is not unambiguously determined without reference to the hyperplane in question. If I had a measurement interaction here, which is now supposed to be represented by that erasure,

22:30 Then on the black hyperplane, I expect no state function to exist because coupling to the apparatus is going on. On the blue hyperplane, I do expect a state function to be assigned and evaluable at that point and not to depend upon the result of the measurement, and on the green hyperplane, a state function will be assigned and evaluable at that point and will depend upon the result of the measurement. Now, this indicates that what one has in the past tended to refer to as a local property of a single quantum mechanical particle at a given Minkowski spacetime point is itself a phrase that is inadequately specified. There is no such thing in general. One must specify at least the hyperplane passing through the point in order to unambiguously pin down the measurement result. On the green hyperplane, every frame of reference assigns a state to that hyperplane. What we had seen before was that in the green frame of reference there was a period of time during which it seemed one had to assign this product state, and there was no corresponding period of time in the black frame of reference in which you had such an intermediate state. It was a one-step process going from the entangled initial state to the final product state, but in the green frame of reference there was an intermediate state. We see now that the reason why there was no time in the black frame during which that intermediate state was assigned was because in the black frame, the hyperplanes to which that intermediate state belongs are not instantaneous in appearance. They have no definite time coordinate. But if one adopts the notion that one must assign state vectors to hyperplanes of all orientations in the presence of measurement, then we see that the green frame of reference product state that is the intermediate state is the unitary transfer form from the black frame to the green frame of a corresponding product state which is associated by the black frame of reference as well as by the green frame of reference, as well, I might say, by the blue frame of reference.

25:00 There is an intermediate state assignment made in all frames of reference to hyperplanes of that orientation lying between these two dashed blue lines, and in every case the assignment made in a given frame is the appropriate unitary transformation of the corresponding assignment. So the whole problem of where do the unitary transforms of the various states that have to be assigned in certain intermediate time intervals in certain frames but for which there seem to be no corresponding time interval in other frames is resolved in an almost trivial manner by generalizing the concept of state vector assignment from state vector assignment at a definite time to state vector assignment on arbitrarily oriented space-like hypotenuse. There are sometimes difficult consistency conditions. These are the two things that must be addressed. Consider a situation in which we have not one measurement interaction taking place, but three, and in this diagram R1 and R2, those two measurement interactions are obviously space-like separated, R3 is partially space-like, partially time-like separated with respect to R1 and R2. I could have moved R3 much further along so that it would be clearly time-like separated with respect to R1 and R2, but I deliberately wanted not to do that. So as to indicate some of the peculiarities of MRIs, I'm going to presume that the quantum mechanical system of interest was initially in the state psi, and remember, that's the state that I would assign not only to hyperplanes of that orientation, but hyperplanes, space-like hyperplanes of any orientation that happened to lie prior to all of the measurement interaction regions. On this hyperplane, which passes through two of the measurement interaction regions, there is no state vector assigned because the system is coupled to two of the apparatus. Here is a hyperplane that lies prior to R1 but after R2. So on that hyperplane, I must assign a state vector which is describing for me the result that emerged from the measurement interaction region R2, but does not depend upon the measurement interaction that took place in R1.

27:30 So I assign a psi j2, j indicating the result, 2 indicating the measurement interaction. Prior to R2, but later than R1, I must assign a state vector that carries the index indicating the result of R1 and the superscript. Now, if I then take this hyperplane and watch the dynamical evolution of the system as this hyperplane changes into that one, or if I start on this hyperplane and watch the dynamical evolution as that hyperplane changes into this one, I find that there are these two very different paths from which I have come from the initial hyperplane All of these terms are related to the initial hyperplane, first to that and then to that, or the initial hyperplane, first to that and then to that, and since the final hyperplane is the same in these two paths, I must get the same state vector assignment, so there is a compatibility condition. It must be possible for me to go from this state to that state to, what is it, psi j i to one, two sequents followed by one, two measurements followed by one measurement, Those two state vectors must be the same state vector, and there is a necessary compatibility on the probabilities that that will happen in either of the two ways of taking the path. Those consistency conditions can, in certain circumstances, be different. And they can indicate that our conception of the necessary commutability of space-like separated measurements may be unnecessarily restrictive. In particular, if we are talking about the kind of measurements that the local field, the local axiomatic field theorists, In which, for a given measurement interaction region, the quantity being measured is represented by some functional or global fields confined to that region for their support. Then you must, and the same is going to appear, then you expect on the basis of Einstein's micro-fasciality for local field theory that these two measurement systems must necessarily commute, that you would have to have the ability, and you can.

30:00 But, in fact, if you define the measurement interaction regions in terms of particle-like observables, such as projection operators on the particle position eigenvectors with the certain... There is neither no necessity, and in many explicit instances, no existence of commutability of the observables and the compatibility conditions cannot necessarily be satisfied, and in those cases, as far as I can see, the only conclusion that one can draw is that you cannot carry out both of those measurements. This is the analog of saying that in ordinary quantum measurement theory, You cannot simultaneously perform with arbitrary precision non-commuting measurements, and you can't use that, you can't say you cannot simultaneously perform here because simultaneity is dependent upon the frame of reference you're in, you simply have to say that in the case in which R2 did not commute with R1, you simply could not perform both R2 and R1 as indicated with arbitrary precision. A counterintuitive result, but not incompatible as far as I can see with anything we've presently done. Space-like hyperplanes are parametrized by a time-like four vector, eta mu, and a tau parameter, tau, such that all the points lying on the space-like hyperplane satisfy this simple linear homogeneous algebraic equation, eta mu, x mu, v v. Under a Lorentz transformation, in homogeneous Lorentz transformation, in which the coordinates of the reference frame go from x to x prime, with homogeneous coefficients lambda and homogeneous parameters a, The time-like unit vector characterizing the orientation of the hyperplane is transformed into eta to the prime, and the tau parameter indicating the location of the hyperplane once its orientation is pinned down goes from tau to tau prime, the difference being given by the product. Any two hyperplanes will be characterized by two sets of parameters, eta one tau one for one hyperplane, eta two tau two for another. Timeline vectors eta1 and eta2 are the same, that means the two hyperplanes are parallel and they will not intersect. If the two eta vectors are distinct, the hyperplanes are non-parallel and they will intersect.

32:30 The algebra of mechanical particles is solved very easily in the context of this formalism. Before I get on, finally, to the relatively new material, the position operators. Suppose we have a three-vector position operator that is time-dependent for a single quantum mechanical particle, and we consider combining it with what we would naively want to call its time component, ct. There is a well-known difficulty in trying to implement anything like a Lorentz transformation on such a piece, for the simple reason that in Lorentz transforms, space and time coordinates are mixed up, so that if I were to try and calculate In the new coordinate system, I would find that the Lorentz transformation mixed the operator value x in with the c-number value ct, and I would have an operator value t', which in that kind of a formalism would not make any sense. However, this is rather trivially resolved by recognizing x and ct as a special case of instantaneous hyperplanes. These are the four-vector position operators associated with an arbitrarily oriented and arbitrarily located hyperplane. This four-vector position operator does not have all four of its components independent. It must satisfy the same constraint, it must satisfy the same equation as defines the hyperplane question itself, so as to guarantee that the expectation values of the four components of x will indeed be associated with the point lying on the e to the tau vector. In the particular case of an instantaneous hyperplane of the eta of kappa 0, 0, this x mu of eta 0 becomes the value of the spatial x.

35:00 The covariant way is the expectation value of the position operator on the eta tau hyperplane in the original frame. This is the expectation value of the position operator on the same hyperplane as it appears in the new frame of reference. In the new frame of reference, it appears to have a different orientation and location than it did in the original frame of reference, but as a consequence of that very change in the mode of description, we are talking about the same hyperplane, and that was crucial for it to solve the purely kinematic that we now get on from the material. Consider a measurement interaction region and think of one hyperplane that is prior to the measurement interaction region on which the system As a state vector psi, and two parallel hyperplanes, there being parallel is not so crucial, but it simplifies the discussion a bit, two parallel hyperplanes, both of which lie later than the measurement interaction region, and have intersection points at least x1u and x2u. And I'm now going to assume that this is a field theoretic system, quantum field theoretic system. Those black lines coming off are supposed to represent the envelope of the forward light cones. Now, there are two kinds of measurements that can take place inside that measurement interaction region, and they're just the two kinds that I talked about before. Suppose I wanted to calculate what the field theorist calls the two-point function at those two points, x1 mu, x2 mu, but not in the vacuum state, but in the actual state that is indicated as belonging to the blue hyperplane prior to the measurement interaction region. The expectation value of the product of the fields at x1 mu and x2 mu. See that? My claim is that in what I'm calling a field-type measurement, in which the object being measured is a functional of the local fields confined to the region bounded by the red circle,

37:30 then I can expect that since the two points lie space-like with respect to all the points inside the region, The expectation value of the two-point product of the fields in the final state, psi sub i, which is quite a different state vector because the measurement is taking place, will be the same as the expectation value of the two-point product of the field in the initial state. But if in fact the measurement is of the other kind, the measurement of a particle like observable, then that equality no longer holds. That's indicative of the kind of measurement that I'm now calling a particle-like measurement, in which all the particle numbers are conserved. Then I find that, in fact, the expectation value of the two field operators is not the same for any one of the possible states resulting from the measurement, and in fact it's also not the same in general. For the statistical sum over all the possible resulting states emerging from the measurement, weighted by their respective probabilities also emerging. I should indicate that in these particle type measurements, it's not always going to be the case that the two are the same quality. There are instances and cases in which the state only contains... If the state were to contain two quanta of the field of the type as we've seen, one might get equality here. What this is suggesting is that you cannot unequivocally pin down the value of these expectation values of endpoint functions of fields by simply stipulating the point at which you evaluate the field. You must also, in the presence of measurement interaction regions, indicate the orientation of the hyperplane passing through the points, because otherwise you can't determine which state vector should be used in the calculation of the expectation value. Could you not take it off and just stand back? Let me apply this idea to EPR. In the case of EPR...

40:00 Here are the two photons, here are the two polarizers. These polarizers, of course, have whorled tubes, which in the black flat part frame are moving straight up the drawing of the whorled tube that indicated that they were spatially static. In this EPR experiment, I have the two photons going out, and I look over here and I pick for myself four points, one, two, three, and four, and I decide to calculate the four-point function. For really what I want to do is the four-point function of the electromagnetic field. I'm not going to worry about the spin of the photons, even though that's quite relevant to the polarizers, I'm going to suppress that in the algebra, and I have a product of four fields here. The initial state is psi, and I can do that calculation. And when I do that calculation, I am referring to the four-point function for hyperplanes of that orientation, because it is on such hyperplanes that the appropriate state vector is the initial state vector, the entangled state vector. I get one result. If I do the same calculations, same in the sense that I'm evaluating the fields of their product on the same four space-time points, but in fact do the evaluation using the state vector that is associated with these hydroponics, a state vector that carries the information of what happened when this photon went through that polarizer, but does not yet carry information as to what happens when that photon went through that polarizer. Then I get a very different result. And I know that because of the nature of this state that the probability of this photon either getting through or not getting through is one-half in both cases. And you can do this calculation and you find that there is an inequality. This is an example of a situation in which if I had simply calculated the two-point function, I would have found equality. By virtue of it being a two-photon state, I don't notice any difference when I calculate a two-point function. But for a four-point function, there is a definite difference which can be calculated. So once again, in the presence of measurements, where in the Heisenberg picture the state vector assignments to the various hyperplanes will differ from hyperplane to hyperplane, the calculation of expectation values of n parts of local fields depends not only on the specification of the points in the Minkowski space to evaluate the fields, but also on the orientation.

42:30 Now comes the punchline. But these two examples, first of all, the manifestly covariant treatment of relativistic quantum mechanical particle position, which obviously depends upon hyperplanes, and could be further elaborated by concentrating on the properties of the eigenvectors of that eigenvalue, where once again it happens in that case and in this case where certain field theoretic functions associated with otherwise purely local fields require reference to the orientation of the hyperplanes and not merely Both of these suggest that in relativistic quantum theory, at least in the presence of measurements, it is not sufficient to think of the underlying substratum from which the basic dynamical variables are defined as simply being points in Minkowski space, but as rather being points on space like hyperplanes in Minkowski space, so that if I have any two points on hyperplanes, Such that their Minkowski coordinates in a given frame of reference are the same, I cannot for physical purposes think of them as the same point because they may be associated with differently oriented hyperplanes passing through those otherwise identical points and that can make fundamental physical difference in the evaluation of fundamental physical quantities. The suggestion then is... From the point of view of the questions of the ontology of space-time, that in some sense, quite prior to anything like the Luzer-Klein theories or supersymmetric theories or any of that novel jargon of the current day, that at the microscopic level and all of this quantum theory, we live in a seven-dimensional, and for a seven-dimensional man, those local structures can refer to, in the theory, we must specify seven parameters in which that idea seems to yield advantages.

45:00 In the attempt to do quantum electrodynamics, we can either proceed in what is called the Coulomb gauge, in which every step of the quantization procedure is susceptible to a clear-cut physical interpretation, in which the Hilbert space of states is positive definite with a direct probability interpretation, but in which the price we pay is the loss of manifest covariance of the formulation. And the demonstration of covariance compatibility with the principle of relativity is awkward, if not difficult, in that formulation. On the other hand, we can quantize electrodynamics in any one of the several covariant gauges, such as the Lorentz gauge, in which the formalism is at every stage manifestly covariant, but the physical interpretation of at least the individual steps in the process of quantization is obscure at best. Where we have to introduce negative and zero-norm states into the Hilbert space, we lose a clean probability interpretation. Some of the field equations must be satisfied as weak constraint equations confined to a subspace of the Hilbert space, etc. I contend that if one is willing to introduce hyperplane dependence in the formulation of electrodynamics, then quantization can be carried out in a manner which is both manifestly covariant