The Dreams That Stuff is Made of
Page 94
We may look at the question also from the point of view of the action principle. If I is the action integral, then
(3-4)
because L is homogeneous of the first degree in the dq/dt. So we can express the action integral with respect to τ in the same form as with respect to t. That shows that the equations of motion which follow from the action principle must be invariant under the passage from t to τ. The equations of motion do not refer to any absolute time.
We have thus a special form of Hamiltonian theory, but in fact this form is not really so special because, starting with any Hamiltonian, it is always permissible to take the time variable as an extra coordinate and bring the theory into a form in which the Hamiltonian is weakly equal to zero. The general rule for doing this is the following: we take t and put it equal to another dynamical coordinate q0. We set up a new Lagrangian
(3-5)
L* involves one more degree of freedom than the original L. L* is not equal to L but
Thus the action is the same whether it refers to L* and τ or to L and t. So for any dynamical system we can treat the time as an extra coordinate q0 and then pass to a new Lagrangian L*, involving one extra degree of freedom and homogeneous of the first degree in the velocities. L* gives us a Hamiltonian which is weakly equal to zero.
This special case of the Hamiltonian formalism where the Hamiltonian is weakly equal to zero is what we need for a relativistic theory, because in a relativistic theory we don’t want to have one particular time playing a special role; we want to have the possibility of various times τ which are all on the same footing. Let us see in detail how we can apply this idea.
FIG. 1
We want to consider states at specified times with respect to different observers. Now if we set up a space-time picture as in Fig. 1, the state at a certain time refers to the physical conditions on a three-dimensional flat space-like surface S1 which is orthogonal to the time axis. The state at different times will refer to physical conditions on different surfaces S2, S3, ... Now we want to bring in other time axes referring to different observers and the state, with respect to the other time axes, will involve physical conditions on other flat space-like surfaces like . We want to have a Hamiltonian theory which will enable us to pass from the state, S1 say, to the state . Starting off with given initial conditions on the surface and applying the equations of motion, we must be able to pass over to the physical conditions on the surface . There must thus be four freedoms in the motion of a state, one freedom corresponding to the movement of the surface parallel to itself, then three more freedoms corresponding to a general change of direction of this flat surface. That means that there will be four arbitrary functions occurring in the solution of the equations of motion which we are trying to get. So we need a Hamiltonian theory with (at least) four primary first-class constraints.
There may be other primary first-class constraints if there are other kinds of freedom in the motion, for example, if we have the possibility of the gauge transformations of electrodynamics. To simplify the discussion, I will ignore this possibility of other first-class primary constraints, and consider only the ones which arise from the requirements of relativity.
We could proceed to set up our theory referring to these flat space-like surfaces which can move with the four freedoms, but I would like first to consider a more general theory in which we consider a state to be defined on an arbitrary curved space-like surface, such as S of Fig. 2. This represents a three-dimensional surface in space-time which has the property of being everywhere space-like, that is to say, the normal to the surface must lie within the light-cone. We may set up a Hamiltonian theory which tells us how the physical conditions vary when we go from one of the curved space-like surfaces to a neighboring one.
Now, bringing in the curved surfaces means bringing in something which is not necessary from the point of view of special relativity. If we wanted to bring in general relativity and gravitational fields, then it would be essential to work with these curved surfaces, but for special relativity, the curved surfaces are not essential. However, I like to bring them in at this stage, even for the discussion of a theory in special relativity, because I find it easier to explain the basic ideas with reference to these curved surfaces than with reference to the flat surfaces. The reason is that with these curved surfaces we can make local deformations of the surface like δS in Fig. 2, and discuss the equations of motion with respect to these local deformations of the surface.
FIG. 2
One way of proceeding now would be to refer our action integral to a set of curved surfaces, like S, take the amount of action between two neighboring curved surfaces, divide it by some parameter δτ expressing the distance between these two surfaces, take this amount of action as our Lagrangian, then apply our standard method of passing from the Lagrangian to the Hamiltonian. Our Lagrangian would necessarily be homogeneous of the first degree in the velocities with respect to the time parameter τ which specifies the passage from one of these space-like surfaces to a neighboring one, and it would lead to a Hamiltonian theory for which the Hamiltonian is weakly equal to zero.
However, I don’t want to go through all the work of following through in detail what we get from an action principle. I want to short-circuit that work and discuss the form of the final Hamiltonian theory which results. We can get quite a lot of information about the form of this Hamiltonian theory just from our knowledge that there must be freedom for the space-like surface to move arbitrarily so long as it remains always space-like. This freedom of motion of the space-like surface must correspond to first-class primary constraints in the Hamiltonian, there being one primary first-class constraint for each type of elementary motion of the surface which can be set up. I shall develop the theory from that point of view.
First of all we have to introduce suitable dynamical variables. Let us describe a point on the space-like surface S by three curvilinear coordinates (x1, x2, x3) = (xr). In order to fix the position of this space-like surface in space-time, we introduce another set of coordinates yΛ(Λ = 0, 1, 2, 3), which we may take to be rectilinear, orthogonal coordinates in special relativity. (I use a capital suffix for referring to the y coordinate system and a small suffix such as r for referring to the x coordinate system.) The four functions yΛ, of xr, will specify the surface S in space time and will also specify its parameterization, i.e. the system of coordinates x1, x2, x3.
We can use these yΛ as dynamical coordinates, q’s. If we form
(3-6)
this is a function of the q’s, the dynamical coordinates.
(3-7)
τ being the parameter changing from one surface to the neighboring surface, will be a velocity, a . Thus yΛ are the dynamical coordinates needed for describing the surface and ẏΛ are the velocities.
We shall need to introduce momentum variables wΛ conjugate to these dynamical coordinates. The momentum variables will be connected with the coordinates by the Poisson bracket relations
(3-8)
We shall need other variables for describing any physical fields which occur in the problem. If we are dealing with a scalar field V, then V(x) for all values of x1, x2, x3 will provide us with further dynamical coordinates, q’s. V, r will be functions of the q’s. ∂V/∂τ will be a velocity. The derivative of V in any direction is expressible of terms of ∂V/∂τ of and V,r and so is expressible in terms of the dynamical coordinates and velocities. The Lagrangian will involve these V’s differentiated in general directions and is thus a function of the dynamical coordinates and velocities. For each V, we shall need a conjugate momentum U, satisfying the Poisson bracket conditions
(3-9)
That is how one would treat a scalar field. There is a similar method for vector, tensor, or spinor fields, just bringing in the necessary additional suffixes. I need not go into that.
Now let us see what the Hamiltonian will be like. The Hamiltonian has to be a linear function of primary first-class constraints of the type (3
-2). First of all I shall put down what the primary first-class constraints are like. There must be primary first-class constraints which allow for arbitrary deformations of the surface. They must involve the variables w to which are conjugate to the y’s, in order to make the y’s vary, and they will involve other field quantities. We can express them in the form
(3-10)
where KΛ is some function of the Hamiltonian variables, the q’s and p’s, not involving the w ’s.
Now we can assert that the Hamiltonian is just an arbitrary linear function of all the quantities (3-10):
(3-11)
This is integrated over the three x’s which specify a point on the surface. The c’s are arbitrary functions of the three x’s and the time.
The general equation of motion is of course ġ ≈ [g, HT ], We can get a meaning for the coefficient cΛ by taking this equation of motion and applying it for g equal to one of the y variables. For g = yΛ at some particular point x1, x2, x3 we get
(3-12)
Here the attached to a field quantity cΓ, wΓ, or KΓ denotes the value of that quantity at the point X1′, X2′, X3′. yΛ has zero Poisson brackets with because is independent of the w’s, so we just have to take into account the Poisson bracket of yΛ with = wΓ(x′). This gives us the delta function and so
(3-13)
Thus the coefficients cΛ turn out to be the velocity variables which tell us how our surface varies with the parameter τ. We can get an arbitrary variation of the surface with τ by choosing these cΛ in an arbitrary way.
This tells us what the Hamiltonian is like for a field theory expressed with respect to states on curvilinear surfaces.
We can make a deeper analysis of this Hamiltonian by resolving the vectors which occur in it into components which are normal and tangential to the surface. If we have any vector whatever, ξΛ, we can obtain from ξΛ a normal component
where lΛ is the unit normal vector, and tangential components (referred to the x coordinate system)
The l are determined by the and are thus functions of the dynamical coordinates. Any vector can be resolved in this way into a part normal to the surface and a part tangential to the surface. We have the scalar product law
(3-14)
where γrs dxr dxs is the metric in the surface referred to the x-coordinate system. γrs is the reciprocal matrix of the γrs. (r, s = 1, 2, 3).
We can use this scalar product law (3-14) to express our total Hamiltonian in terms of the tangential and normal components of w and K:
(3-15)
Here ẏ = ẏΛlΛ and ẏr = ẏΛyΛ,r.
We shall need the Poisson bracket relationships between the normal and tangential terms in (3-15). I will first write down the Poisson bracket relations for the different components of w. We have of course
(3-16)
referred to the external coordinates y ; but when we resolve our w’s into normal and tangential components, they will no longer have zero Poisson brackets with each other. The Poisson brackets can easily be worked out by straightforward arguments. I don’t want to go through the details of that work. I will just mention that the details can be found in a paper of mine (Canadian Journal of Mathematics, 3, 1 (1951)). The results are
(3-17)
(3-18)
(3-19)
Now we know ] that
(3-20)
We can infer that
(3-21)
(3-22)
(3-23)
These results could be worked out directly from the definitions of the normal and tangential components of the w’s, but they can be inferred more simply by the following argument. Since wr + Kr, w⊥ + K⊥ are all first class, their Poisson brackets are zero weakly. Thus , and [w⊥ + K⊥, + must all be weakly equal to zero. We can now infer what they are equal to strongly. We have to put on the right-hand side in each of (3-21,) (3-22,) and (3-23) a quantity which is weakly equal to zero and which is therefore built up from wr + Kr and w⊥ + K⊥ with certain coefficients. Further, we can see what these coefficients are by working out what terms containing w there are on the right-hand sides. Terms containing w’s can arise only from taking the Poisson bracket of a w with a w, according to (3-17), (3-18), and (3-19). Taking a Poisson bracket [w, K′] will not lead to anything involving w, because it means taking the Poisson bracket of a w momentum with some functions of dynamical coordinates and momenta other than w’s, and that won’t involve the w momentum variables. Similarly the Poisson bracket of a K with a won’t involve any w variables. Thus the only to variables which occur on the right side of (3-21) will be the ones which occur on the right side of (3-17). We have to put certain further terms in the right side of (3-21) in order that the total expression shall be weakly equal to zero. It is then quite clear what we should put here, namely, (ws + Ks)δ,r (x − x′) + δ,s (x − x’). We do the same with the right sides of (3-22), (3-23).
The next thing to notice is that the terms ws + Ks in the Hamiltonian (3-15) correspond to a motion in which we change the system of coordinates in the curved surface but do not have the surface moving. It corresponds to each point in the surface moving tangentially to the surface.
Let us put ẏ⊥ = 0, which means that we are taking no motion of the surface perpendicular to itself but are merely making a change of the coordinates of the surface, and then we have equations of motion of the type
(3-24)
This must be the equation of motion which tells us how g varies when we change the system of coordinates in the surface without moving the surface itself. Now this change in g must be a trivial one, which can be inferred merely from the geometrical nature of the dynamical variable g. If g is a scalar, then we know how that changes when we change the system of coordinates x1, x2, x3. If it is a component of a vector or a tensor there will be a rather more complicated change for g, but still we can work it out; similarly if g is a spinor. In every case, this change of g is a trivial one. That means that Ks, can be determined from geometrical arguments only.
I will give one or two examples of that. For a scalar field V with a conjugate momentum U, there is a term
(3-25)
in Kr. For a vector field, say a three-vector As, with conjugate Bs, there is a term
(3-26)
in Kr; and so on for tensors, with something rather more complicated for spinors. The first term in (3-26) is the change in As coming from the translation associated with the change in the system of coordinates, and the second is the change in the As arising from the rotation associated with the change in the system of coordinates. There is no such rotation term coming in in the case (3-25) of the scalar.
We can obtain the total Kr by adding the contribution needed for all the different kinds of fields which are present in the problem. The result is that we can work out this tangential component of K just from geometrical arguments. One can see in this way that the tangential component of K is something which is not of real physical importance, it is just concerned with the mathematical technique. The quantity which is of real physical importance is the normal component of K in (3-15). This normal component of K added on to the normal component of w gives us the first-class constraint which is associated with a motion of the surface normal to itself. That is something which is of dynamical importance.
The problem of getting a Hamiltonian field theory on these curved surfaces involves finding the expressions K to satisfy the required Poisson bracket relations (3-21), (3-22), and (3-23). The tangential part of K can be worked out from geometrical arguments as I discussed, and when we have worked it out we should find of course that it satisfies the first Poisson bracket relation (3-21). The second Poisson bracket relation (3-22) involves K⊥ linearly and this Poisson bracket relation would be satisfied by any quantity K⊥ which satisfies the condition of being a scalar density. This Poisson bracket relation really tells us that if the quantity K⊥ varies suitably under a change of coordinate system X1, X2, X3, this Poisson bracket relation will be fulfilled. The difficult relation
to fulfill is the third one, which is quadratic in K⊥. So the problem of setting up a Hamiltonian field theory on curved space-like surfaces is reduced to the problem of finding a normal component of K which is a scalar density and which satisfies the Poisson bracket relationship (3-23).
One way of finding such a normal component of K is to work from a Lorentzinvariant action principle. We might obtain all the components of K by working from the action principle. If we did that, the tangential part of K which we get would not necessarily be the same as that built up from terms like (3-25) and (3-26), because it might differ by a contact transformation. But one could eliminate such a contact transformation by rewriting the action principle, adding to it a perfect differential term. This doesn’t affect the equations of motion. By such a change of the action principle, one can arrange that the tangential part of K given by the action principle agrees precisely with the value which is obtained by the simple application of geometrical arguments. We are then able to find the normal component of K by working with our general method of passing from the action principle to the Hamiltonian. If the action principle is relativistic, then the normal component of K obtained in this way would have to satisfy the condition (3-23).