(1-37)
The coefficients v are completely arbitrary and at our disposal. Suppose we take different values, v, for these coefficients. That would give a different g(δt), the difference being
(1-38)
We may write this as
(1-39)
where
(1-40)
is a small arbitrary number, small because of the coefficient δt and arbitrary because the v’s and the v’s are arbitrary. We can change all our Hamiltonian variables in accordance with the rule (1-39) and the new Hamiltonian variables will describe the same state. This change in the Hamiltonian variables consists in applying an infinitesimal contact transformation with a generating function εa φ a . We come to the conclusion that the φa’s, which appeared in the theory in the first place as the primary first-class constraints, have this meaning: as generating functions of infinitesimal contact transformations, they lead to changes in the q’s and the p’s that do not affect the physical state.
However, that is not the end of the story. We can go on further in the same direction. Suppose we apply two of these contact transformations in succession. Apply first a contact transformation with generating function εa φ a and then apply a second contact transformation with generating function γ a φ a , where the gamma’s are some new small coefficients. We get finally
(1-41)
(I retain the second order terms involving products εγ , but I neglect the second order terms involving ε2 or involving γ 2 . This is legitimate and sufficient. I do that because I do not want to write down more than I really need for getting the desired result.) If we apply the two transformations in succession in the reverse order, w get finally
(1-42)
Now let us subtract these two. The difference is
(1-43)
By Jacobi’s identity (1-13) this reduces to
. (1-44)
This Δg must also correspond to a change in the q’s and the p’s which does not involve any change in the physical state, because it is made up by processes which individually don’t involve any change in the physical state. Thus we see that we can use
(1-45)
as a generating function of an infinitesimal contact transformation and it will still cause no change in the physical state.
Now the φa are first-class: their Poisson brackets are weakly zero, and therefore strongly equal to some linear function of the φ’s. This linear function of the φ’s must be first-class because of the theorem I proved a little while back, that the Poisson bracket of two first-class quantities is first-class. So we see that the transformations which we get this way, corresponding to no change in the physical state, are transformations for which the generating function is a first-class constraint. The only way these transformations are more general than the ones we had before is that the generating functions which we had before are restricted to be first-class primary constraints. Those that we get now could be first-class secondary constraints. The result of this calculation is to show that we might have a first-class secondary constraint as a generating function of an infinitesimal contact transformation which leads to a change in the q’s and the p’s without changing the state.
For the sake of completeness, there is a little bit of further work one ought to do which shows that a Poisson bracket [H , φa ] of the first-class Hamiltonian H’ with a first-class φ is again a linear function of first-class constraints. This can also be shown to be a possible generator for infinitesimal contact transformations which do not change the state.
The final result is that those transformations of the dynamical variables which do not change physical states are infinitesimal contact transformations in which the generating function is a primary first-class constraint or possibly a secondary first-class constraint. A good many of the secondary first-class constraints do turn up by the process (1-45) or as [H, φ a ]. I think it may be that all the first-class secondary constraints should be included among the transformations which don’t change the physical state, but I haven’t been able to prove it. Also, I haven’t found any example for which there exist first-class secondary constraints which do generate a change in the physical state.
DR. DIRAC
Lecture No. 2
THE PROBLEM OF QUANTIZATION
We were led to the idea that there are certain changes in the p’s and q’s that do not correspond to a change of state, and which have as generators first-class secondary constraints. That suggests that one should generalize the equations of motion in order to allow as variation of a dynamical variable g with the time not only any variation given by (1-21), but also any variation which does not correspond to a change of state. So we should consider a more general equation of motion
(2-1)
with an extended Hamiltonian HE, consisting of the previous Hamiltonian, HT , plus all those generators which do not change the state, with arbitrary coefficients:
(2-2)
Those generators φa , which are not included already in HT will be the first-class secondary constraints. The presence of these further terms in the Hamiltonian will give further changes in g, but these further changes in g do not correspond to any change of state and so they should certainly be included, even though we did not arrive at these further changes of g by direct work from the Lagrangian.
That, then, is the general Hamiltonian theory. The theory as I have developed it applies to a finite number of degrees of freedom but we can easily extend it to the case of an infinite number of degrees of freedom. Our suffix denoting the degree of freedom is n = 1, . . . , N ; we may easily make N infinite. We may further generalize it by allowing the number of degrees of freedom to be continuously infinite. That is to say, we may have as our q’s and p’s variables qx , px where x is a suffix which can take on all values in a continuous range. If we work with this continuous x, then we have to change all our sums over n in the previous work into integrals. The previous work can all be taken over directly with this change.
There is just one equation which we will have to think of a bit differently, the equation which defines the momentum variables,
(1-3)
If n takes on a continuous range of values, we have to understand by this partial differentiation a process of partial functional differentiation that can be made precise in this way: We vary the velocities by δx in the Lagrangian and then put
(2-3)
The coefficient of δx occurring in the integrand in δL is defined to be px.
After giving this general abstract theory, I think it would be a help if I gave a simple example as illustration. I will take as an example just the electromagnetic field of Maxwell, which is defined in terms of potentials Aµ. The dynamical coordinates now consist of the potentials for all points of space at a certain time. That is to say, the dynamical coordinates consist of Aµx , where the suffix x stands for the three coordinates x1, x2, x3 of a point in three-dimensional space at a certain time x0 (not the four x’s which one is used to in relativity). We shall have then as the dynamical velocities the time derivatives of the dynamical coordinates, and I shall denote these by a suffix 0 preceded by a comma.
Any suffix with a comma before it denotes differentiation according to the general scheme
(2-4)
We are dealing with special relativity so that we can raise and lower these suffixes according to the rules of special relativity: we have a change in sign if we raise or lower a suffix 1, 2, or 3 but no change of sign when we raise or lower the suffix 0.
We have as our Lagrangian for the Maxwell electrodynamics, if we work in Heaviside units,
(2-5)
Here d3 x means dx1 dx2 dx3 , the integration is over three-dimensional space, and Fµν means the field quantities defined in terms of the potentials by
(2-6)
This L is the Lagrangian because its time integral is the action integral of the Maxwell field.
Let us now take this Lagrangian and apply the rules of our formalism for passing to the Hamiltonian. W
e first of all have to introduce the momenta. We do that by varying the velocities in the Lagrangian. If we vary the velocities, we have
(2-7)
Now the momenta Bµ are defined by
(2-8)
and these momenta will satisfy the basic Poisson bracket relations
(2-9)
In this formula A is taken at a point x in three-dimensional space and B is taken at a point x′ in the three-dimensional space. is just the Kronecker delta function. δ3 (x − x′) is the three-dimensional delta function of x − x′.
We compare the two expressions (2-7) and (2-8) for δL and that gives us
(2-10)
Now Fµν is anti-symmetrical
(2-11)
So if we put µ = 0, in (2-10) we get zero. Thus is equal to zero. This is a primary constraint. I write it as a weak equation:
(2-12)
The other three momenta Br (r = 1, 2, 3) are just equal to the components of the electric field.
I should remind you that (2-12) is not just one primary constraint: there is a whole threefold infinity of primary constraints because there is the suffix x which stands for some point in three-dimensional space; and each value for x will give us a different primary constraint.
Let us now introduce the Hamiltonian. We define that in the usual way by
(2-13)
I’ve done a partial integration of the last term in (2-13) to get it in this form. Now here we have an expression for the Hamiltonian which does not involve any velocities. It involves only dynamical coordinates and momenta. It is true that Frs involves partial differentiations of the potentials, but it involves partial differentiations only with respect to x1, x2, x3. That does not bring in any velocities. These partial derivatives are functions of the dynamical coordinates.
We can now work out the consistency conditions by using the primary constraints (2-12). Since they have to remain satisfied at all times, [B0, H] has to be zero. This leads to the equation
(2-14)
This is again a constraint because there are no velocities occurring in it. This is a secondary constraint, which appears in the Maxwell theory in this way. If we proceed further to examine the consistency relations, we must work out
(2-15)
We find that this reduces to 0 = 0.It does not give us anything new, but is automatically satisfied. We have therefore obtained all the constraints in our problem. (2-12) gives the primary constraints. (2-14) gives the secondary constraints.
We now have to look to see whether they are first-class or second-class, and we easily see that they are all first-class. The B0 are momenta variables. They all have zero Poisson brackets with each other. and B0 0 also have zero Poisson brackets with each other. And and B,r x also have zero Poisson brackets with each other. All these quantities are therefore first-class constraints. There are no second-class constraints occurring in the Maxwell electrodynamics.
The expression (2-13) for H is first-class, so this H can be taken as the H′ of (1-33). Let us now see what the total Hamiltonian is:
(2-16)
This v x is an arbitrary coefficient for each point in three-dimensional space. We have just added on the primary first-class constraints with arbitrary coefficients, which is what we must do according to the rules to get the total Hamiltonian.
In terms of the total Hamiltonian we have the equation of motion in the standard form
(1-21)
The g which we have here may be any field quantity at some point x in three-dimensional space, or may also be a function of field quantities at different points in three-dimensional space. It could, for example, be an integral over three-dimensional space. This g can be perfectly generally any function of the q’s and the p’s throughout three-dimensional space.
It is permissible to take g = A0 and then we get
(2-17)
because A0 has zero Poisson brackets with everything except the B0 occurring in the last term of (2-16). This gives us a meaning for the arbitrary coefficient vx occurring in the total Hamiltonian. It is the time derivative of A0.
Now to get the most general motion which is physically permissible, we ought to pass over to the extended Hamiltonian. To do this we add on the first-class secondary constraints with arbitrary coefficients ux. This gives the extended Hamiltonian:
(2-18)
Bringing in this extra term into the Hamiltonian allows a more general motion. It gives more variation of the q’s and the p’s, of the nature of a gauge transformation. When this additional variation of the q’s and the p’s is brought in, it leads to a further set of q’s and which must correspond to the same state.
That is the result of working out, according to our rules, the Hamiltonian form of the Maxwell theory. When we’ve got to this stage, we see that there is a certain simplification which is possible. This simplification comes about because the variables A0, B0 are not of any physical significance. Let us see what the equations of motion tell us about A0 and B0. B0 = 0 all the time. That is not of interest. A0 is something whose time derivative is quite arbitrary. That again is something which is not of interest. The variables A0 and B0 are therefore not of interest at all. We can drop them out from the theory and that will lead to a simplified Hamiltonian formalism where we have fewer degrees of freedom, but still retain all the degrees of freedom which are physically of interest.
In order to carry out this discard of the variables A0 and B0, we drop out the term νxB0 from the Hamiltonian. This term merely has the effect of allowing A0 to vary arbitrarily. The term − A0,r in HT can be combined with the ux, Br,r in the extended Hamiltonian. The coefficient ux is an arbitrary coefficient in any case. When we combine these two terms, we just have this ux replaced by = ux − A0 which is equally arbitrary. So that we get a new Hamiltonian
(2-19)
This Hamiltonian is sufficient to give the equations of motion for all the variables which are of physical interest. The variables A0, B0 no longer appear in it. This is the Hamiltonian for the Maxwell theory in its simplest form.
Now the usual Hamiltonian which people work with in quantum electrodynamics is not quite the same as that. The usual one is based on a theory which was originally set up by Fermi. Fermi’s theory involves putting this restriction on the potentials:
(2-20)
It is quite permissible to bring in this restriction on the gauge. The Hamiltonian theory which I have given here does not involve this restriction, so that it allows a completely general gauge. It’s thus a somewhat different formalism from the Fermi formalism. It’s a formalism which displays the full transforming power of the Maxwell theory, which we get when we have completely general changes of gauge. This Maxwell theory gives us an illustration of the general ideas of primary and secondary constraints.
I would like now to go back to general theory and to consider the problem of quantizing the Hamiltonian theory. To discuss this question of quantization, let us first take the case when there are no second-class constraints, when all the constraints are first-class. We make our dynamical coordinates and momenta, the q’s and p’s, into operators satisfying commutation relations which correspond to the Poisson bracket relations of the classical theory. That is quite straightforward. Then we set up a Schrödinger equation
(2-21)
ψ is the wave function on which the q’s and the p’a operate. H′ is the first-class Hamiltonian of our theory.
We further impose certain supplementary conditions on the wave function, namely:
(2-22)
Each of our constraints thus leads to a supplementary condition on the wave function. (The constraints, remember, are now all first-class.)
The first thing we have to do now is to see whether these equations for ψ are consistent with one another. Let us take two of the supplementary conditions and see whether they are consistent. Let us take (2-22) and
(2-22′)
If we multiply (2-22) by φj′, we get
(2-23)
If we multiply (2-22)′
by φj, we get
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