The Dreams That Stuff is Made of
Page 96
(4-19)
which expresses that the various translations commute;
(4-20)
and
(4-21)
Let us now consider the requirements for the equations (4-13) and (4-14) to be first-class. The Poisson bracket of any two of them must be something which vanishes weakly and must therefore be a linear combination of them. So we are led to these Poisson bracket relations:
(4-22)
(4-23)
and
(4-24)
The argument for getting these relations is that, on the right-hand sides we had to put something which is weakly equal to zero in each case, and we know the terms on the right-hand sides which involve the momentum variables P, M because these terms come only from the Poisson brackets of momenta with momenta and so are given by (4-19), (4-20), and (4-21). (I have already used the same argument in the curvilinear case for (3-21), (3-22), and (3-23), so there is no need to go into detail here. For example, see how (4-23) comes about. The terms involving P are just the same as in (4-20). They come from the Poisson bracket of P and M. The remaining terms are filled in in order to make the total expression weakly equal to zero.) (4-22), (4-23), and (4-24) are the requirements for consistency.
We can make a further simplification, which we could not do in the case of curvilinear coordinates, in this way: Let us suppose that our basic field quantities are chosen to refer only to the x coordinate system. They are field quantities at specific points x in the surface, and we can choose them so as to be quite independent of the y coordinate system. Then the quantities K⊥, Kr will be quite independent of the y coordinate system, and that means that they will have zero Poisson brackets with the variables P, M. We then have a zero Poisson bracket between each of the variables, p, m and each of the P, M.
This condition follows with the natural choice of dynamical variables to describe the physical fields which are present. We cannot do the corresponding simplification when we are working with the curved surfaces, because the grs variables that fix the metric will enter into the quantities K⊥, Kr. The result is that we cannot set them up in a form which does not refer at all to the y coordinate system, because the y coordinates enter into the grs variables. However, with the flat surfaces, we can make this simplification, and that results in equations (4-22), (4-23), and (4-24) simplifying to
(4-25)
(4-26)
and
(4-27)
P and M have disappeared from these equations, so the consistency conditions now involve only the field variables, and not the variables, which are introduced for describing the surface. In fact, these conditions merely say that the p, m shall satisfy Poisson bracket relations corresponding to the operators of translation and rotation in flat space-time. The problem of setting up a relativistic field theory now reduces to finding the quantities p, m to satisfy the Poisson bracket relations (4-25), (4-26), and (4-27).
These quantities, remember, are defined in terms of K⊥ and Kr, the energy density and the momentum density. The expression for the momentum density is just the same as in curvilinear coordinates. It is determined by geometrical arguments only. Our problem reduces to finding the energy density K⊥ leading to p’s and m’s such that the Poisson bracket relations (4-25), 4-26), and (4-27) are fulfilled.
If we work from a Lorentz-invariant action integral and deduce K⊥ from it by standard Hamiltonian methods, K⊥ will automatically satisfy these requirements in the classical theory. The problem of getting a relativistic quantum theory then reduces the problem of suitably choosing the order of factors which occur in K⊥ so as to satisfy the equations (4-25), (4-26), and (4-27) also in the quantum theory, where the Poisson bracket becomes a commutator and the p, m involve non-commuting quantities.
Let us look at (4-25), (4-26), and (4-27) and substitute for p and m their values in terms of K’s. Then you see that some of these conditions will be independent of K⊥. These are automatically satisfied when we choose Kr properly, in accordance with the geometrical requirements. Some of the conditions are linear in K⊥. These will be satisfied by taking K⊥ to be any three-dimensional scalar density in the space of the x’s. So that there is no problem in satisfying the conditions which are linear in K⊥. The awkward ones to satisfy are the ones which are quadratic in K⊥. They are the following:
(4-28)
(This equation comes from (4-26) where we put µ = ⊥, p = r, and σ = ⊥.)
(4-29)
(from (4-27) where we take ν = ⊥ and σ = ⊥). So the problem of getting a relativistic quantum field theory now reduces to the problem of finding an energy density K⊥ which satisfies the conditions (4-28) and (4-29) when we take into account non-commutation of the factors.
We can analyze these conditions a little more when we take into account that the Poisson bracket connecting ⊥ at one point and at another point will be a sum of terms involving delta functions and derivatives of delta functions:
(4-30)
(This delta is the three-dimensional delta function involving the three coordinates x and the three coordinates x’ of the first and second points.) Here a = a(x), b = b(x), c = c (x), . . . One could have the coefficients involving also x’, but then one could replace them by coefficients involving x only at the expense of making some changes in the earlier coefficients in the series. There is no fundamental dissymmetry between x and x, only a dissymmetry in regard to the way the equation is written.
(4-30) is the general relationship connecting the energy density at two points. Now for many examples, including all the more usual fields, derivatives of the delta function higher than the second do not occur. Let us examine this case further.
Assume derivatives higher that the second do not occur. That means that the series (4-30) stops at the third term. In this special case we can get quite a bit of information about the coefficients a, b, c by making use of the condition that the Poisson bracket (4-30) is anti-symmetrical between the two points x and x’. Interchanging x and x’ in (4-30), we get
(since ∂ br (x)/∂xr = 0, etc.)
(4-31)
The expression (4-31) must equal minus the expression (4-30) identically. In order that the coefficients of δ,rs shall agree we must have
(4-32)
This then makes the coefficients of δ,r agree. Finally, in order that the coefficients of δ shall agree, we must have
(4-33)
This gives us the equation
(4-34)
Let us now substitute in (4-28) and (4-29). They become:
(4-35)
(Note that xr,s = ∂ xr/∂x3 = −δrs .)
(4-36)
This is what our consistency conditions reduce to, and we see that they are satisfied by taking br = Kr. This is not quite the most general solution; more generally we could have
(4-37)
for any quantity θrs satisfying the condition that
(4-38)
Thus θ can have any symmetrical part and its anti-symmetrical part must be a divergence.
That gives the general requirement for a field theory to be relativistic. We have to find the energy density K⊥ satisfying the Poisson bracket relation (4-34) where br is connected with the momentum density by (4-37). If we work out the energy density from a Lorentz-invariant action then this condition will certainly be satisfied in the classical theory. It might not be satisfied in the quantum theory because the order of the factors might be wrong. It is only when one can choose the order of the factors in the energy density so as to make (4-34), (4-37) hold accurately that we have a relativistic quantum theory. The conditions which we have here for a quantum theory to be relativistic are less stringent than the ones which we obtained when we had states defined on general curved surfaces.
I would like to illustrate that by taking the example of Born-Infeld electrodynamics. This is an electrodynamics which is in agreement with Maxwell electrodynamics for weak fields but differs from it for strong fields. (We now refer the electromagnetic field quantit
ies to some absolute unit defined in terms of the charge of the electron and classical radius of the electron, so that we can talk of strong fields and weak fields.) The general equations of the Born-Infeld electrodynamics follow from the action principle:
(4-39)
We may use curvilinear coordinates at this stage. gµν gives the metric referred to these curvilinear coordinates and Fµν gives the electromagnetic field referred to the absolute unit.
We can pass from this action integral to a Hamiltonian by using the general procedure. The result is to give us a Hamiltonian in which we have, in addition to the variables needed to describe the surface, the dynamical coordinates Ar, r = 1, 2, 3. A0 turns out to be an unimportant variable just like in the Maxwell field. The conjugate momenta D r to the Ar are the components of the electric induction, and satisfy the Poisson bracket relations
(4-40)
It turns out that in the Hamiltonian we only have A occurring through its curl, namely through the field quantities:
(4-41)
εrst = 1 when (rst) = (1, 2, 3) and is anti-symmetrical between the suffixes. The commutation relation between B and D is
(4-42)
The momentum density now has the value
(4-43)
This is just the same as in the Maxwell theory. It is in agreement with the general principle that the momentum density depends only on geometrical arguments, i.e. on the geometrical character of the fields we are using, and the action principle doesn’t matter.
The energy density now has the value
(4-44)
Here γrs is the metric in the three-dimensional surface and
(4-45)
If we work with curved surfaces we require K⊥ to satisfy the Poisson bracket relation (3-31). In the classical theory it must do so because it is deduced from a Lorentz-invariant action integral. But we cannot get it to satisfy the required commutation relationship in the quantum theory. The expression for K⊥ has a square root occurring in it, which makes it very awkward to work with. It seems to be quite hopeless to try to get the commutation relations correctly fulfilled with the coefficients γrs occurring on the left. So it does not seem to be possible to get a Born-Infeld quantum electrodynamics with the state defined on general curved surfaces.
Let us, however, go over to flat surfaces. For that purpose, we need to work out the Poisson bracket relationship (4-34). Now we know that conditions are all right in the classical theory. Classically we must therefore have the Poisson bracket relationship:
(4-46)
We can see without going into detailed calculations that this must hold also in the quantum theory, because Kr is built up entirely from the quantities D s and Bt. When we work things out in the quantum theory, we shall have the D’s and the B’s occurring in a certain order, but the D’s and B’s all commute with each other when we take them at the same point. We see that from (4-42). If we put the x’ = x we get
(4-47)
(the derivative of the delta function with the argument 0 is to be taken as zero). Thus we are not bothered by the non-commutation of the D’s and the B’s that occur in Kr. We must therefore get the classical expression, so that the consistency conditions are fulfilled.
So for the Born-Infeld electrodynamics, the consistency conditions for the quantum theory on flat surfaces are fulfilled, while they are not fulfilled on curved surfaces. Physically that means that we can set up the basic equations for a quantum theory of the Born-Infeld electrodynamics agreeing with special relativity, but we should have difficulties if we wanted to have this quantum theory agreeing with general relativity.
That completes the discussion of the consistency requirements for the quantum theory to be relativistic. However, even if we have satisfied these consistency requirements, we have not yet disposed of all the difficulties. There are some quite formidable difficulties still lying ahead of us. If we were dealing with a system involving only a finite number of degrees of freedom, then we should have disposed of all of the difficulties, and it would be a straightforward matter to solve the differential equations on ψ. But with field theory, we have an infinite number of degrees of freedom, and this infinity may lead to trouble. It usually does lead to trouble.
We have to solve equations in which the unknown, the wave function ψ, involves an infinite number of variables. The usual method that people have for solving this kind of equation is to use perturbation methods in which the wave function is expanded in powers of some small parameter, and one tries to get a solution step by step. But one usually runs into the difficulty that after a certain stage the equations lead to divergent integrals.
People have done a great deal of work on this problem. They have found methods for handling these divergent integrals which seem to be tolerable to physicists even though they cannot be justified mathematically, and they have built up the renormalization technique, which allows one to disregard the infinities in the case of certain kinds of field theory.
So, even when we have formally satisfied the consistency requirements, we still have the difficulty that we may not know how to get solutions of the wave equation satisfying the required supplementary conditions. If we can get such solutions, there remains the further problem of introducing scalar products for these solutions, which means considering these solutions as the vectors in a Hilbert space. It is necessary to introduce these scalar products before we can get a physical interpretation for our wave function in terms of the standard rules for the physical interpretation of quantum mechanics. It is necessary that we should have scalar products for the wave functions which satisfy the supplementary conditions, but we do not need to worry about scalar products for general wave functions which do not satisfy the supplementary conditions. There may be no way of defining scalar products for these general wave functions, but that would not matter at all. The physical interpretation for quantum mechanics requires that scalar products exist only for wave functions satisfying all the supplementary conditions.
You see that there are quite formidable difficulties in getting the Hamiltonian theory to work, in connection with quantum mechanics. So far as concerns classical mechanics, the method seems to be fairly complete and we know exactly what the situation is; but for quantum mechanics we have only really started on the problem. There are the difficulties of finding solutions even when the supplementary conditions are formally consistent, and possibly also the difficulty of introducing scalar products of the solutions.
The difficulties are quite serious, and they have led some physicists to challenge the whole Hamiltonian method. A good many physicists are now working on the problem of trying to set up a quantum field theory independently of any Hamiltonian. Their general method is to introduce quantities which are of physical importance, then to bring in accepted general principles in order to impose conditions on these quantities; and their hope is that ultimately they will get enough conditions imposed on these quantities of physical importance to be able to calculate them. They are still very far from achieving that end, and my own belief is that it will not be possible to dispense entirely with the Hamiltonian method. The Hamiltonian method dominates mechanics from the classical point of view. It may be that our method of passing from classical mechanics to quantum mechanics is not yet correct. I still think that in any future quantum theory there will have to be something corresponding to Hamiltonian theory, even if it is not in the same form as at present.
I have given the treatment of the Hamiltonian method as far as it has yet been developed. It is quite a general and powerful method which can be adapted to a variety of problems. It can be adapted to problems where singularities (point or surface) occur in the field. The general idea governing this development of the Hamiltonian theory is to find an action I which involves certain parameters q, such that when we vary the q’s, δI is linear in the δq’s. It is indispensable that we should have δI linear in the δq’s in order that we may apply the treatment described in these lectures.
The way to brin
g about linearity when we have singularities is to work in terms of curvilinear coordinates, and not to vary any equations which determine the position of a singular point or a singular surface. For example, if we are dealing with a singular surface specified by an equation f(x) = 0, then we must have a variation principle in which f(x) is not varied. If we allow f(x) to vary, if we treat f itself as providing some of the q’s, then we do not have δI linear in the δq’s. But we can keep f(x) fixed with respect to some curvilinear coordinate system x and we can vary the surface by varying the curvilinear coordinate system without varying the function f. Then the general method which I have discussed here works very well in the classical theory. When we go over to quantization we have the difficulties arising which I have discussed.