The Dreams That Stuff is Made of
Page 95
We can now discuss the passage to the quantum theory. Quantization involves making the quantities w and the variables which enter in K into operators. We have to be careful now how we define the tangential and the normal components of w, and I choose this way to define them:
(3-27)
putting the momentum variable w on the right. (In the quantum theory, you see, the result is different, depending on whether we put the w on the right or the left.) Similarly,
(3-28)
Then these quantities are well defined.
Now in the quantum theory we have the weak equations wr + Kr ≈ 0 and w⊥ + K⊥ ≈ 0, which provide us with supplementary conditions on the wave function:
(3-29)
(3-30)
corresponding to (2-22). We require that these supplementary conditions be consistent. According to (2-25), we must arrange that in the commutation relations (3-21), (3-22), and (3-23) the coefficients on the right-hand sides stand before (on the left of) the constraints.
In the case of (3-21), the tangential components, the conditions fit if we choose the order of the factors in Kr so that the momentum variables are always on the right. We have now in (3-21) a number of quantities, linear in the momentum variables with the momentum variables on the right, and the commutator of any two such quantities will again be linear in the momentum variables with the momentum variables on the right. Thus we shall always have the momentum variables on the right and we shall always have our factors occurring in the order in which we want them to.
Now we have the problem of bringing in K⊥, which cannot be disposed of so simply. K⊥ will usually involve the product of non-commuting factors and we have to arrange the order of those factors so that (3-22) and (3-23) shall be satisfied with the coefficients occurring on the left in every term on the right-hand side. The equation (3-22) is again a fairly simple one to dispose of. If we simply take K⊥ to be a scalar density, that is all that is needed, because we have w⊥ + K⊥ occurring on the right-hand side without any coefficients which don’t commute with it; the only coefficient is the delta function, which is a number.
But the relationship (3-23) is the troublesome one. For the purposes of the quantum theory, I ought to write out the right-hand side here rather more explicitly:
(3-31)
I’ve written this out with the coefficients γrs occurring on the left, and that is how we need to have these coefficients in the quantum theory.
The problem of setting up a quantum field theory on general curved surfaces involves finding K⊥ so that this Poisson bracket relationship (3-31) holds with the coefficients γrs occurring on the left. If we do satisfy (3-31), then the supplementary conditions (3-30) are consistent with each other, and we already have (3-29) consistent with each other and (3-30) consistent with (3-29).
There we have formulated the conditions for our quantum theory to be relativistic. We need a bit of luck to be able to satisfy the conditions. We cannot always satisfy them. There is one general rule which is of importance, which tells us that when we’ve got a K⊥ satisfying these conditions and certain other conditions, we can easily construct other K⊥’s to satisfy the conditions. Let us suppose that we have a solution in which K⊥ involves only undifferentiated momentum variables together with dynamical coordinates which may be differentiated. There are a number of simple fields for which K⊥ does satisfy the Poisson bracket relations (3-22) and (3-23) and does have this simple character. Then we may add to K⊥ any function of the undifferentiated q’s. That is to say, we take a new K⊥,
Then we see that adding on this φ to K⊥ can affect the right-hand side of (3-31) only by bringing in a multiple of the delta function. We cannot get any differentiations of the delta function coming in, because the extra terms come from Poisson brackets of φ(q) with undifferentiated momentum variables. So that the only effect on the right-hand side of adding the term φ to K⊥ can be adding on a multiple of the delta function. But the right-hand side has to be anti-symmetrical between x and x′, because the left-hand side is obviously antisymmetrical between x and x’. That prevents us from just adding a multiple of the delta function to the right-hand side of (3-31), so that it is not altered at all. Thus if the original K⊥ satisfies the Poisson bracket relation (3-31), then the new one will also satisfy it.
There is a further factor which has to be taken into account to complete the proof. φ may also involve Γ . One finds that [w⊥,Γ′] involves δ(x − x’) undifferentiated (one just has to work this out) and thus we can bring Γ into φ without disturbing the argument. In fact, we have to bring in Γ in order to preserve the validity of (3-22), which requires that and K⊥ shall be scalar ensities. We must thus bring in a suitable power of Γ to make φ a scalar density.
This is the method which is usually used in practice for bringing in interaction between fields without disturbing the relativistic character of the theory. For various simple fields the conditions turn out to be satisfied. We have the necessary bit of luck, and we can bring in interaction between fields of the simple character described and the conditions for the quantum theory to be relativistic are preserved.
There are some examples for which we don’t have the necessary luck and we just cannot arrange the factors in K⊥ to get (3-31) holding with the coefficients on the left, and then we do not know how to quantize the theory with states on curved surfaces. But actually, we are trying to do rather more than is necessary when we try to set up our quantum theory with states on curved surfaces. For the purposes of getting a theory in agreement with special relativity, it would be quite sufficient to have our states defined only on flat surfaces. That will involve some conditions on K⊥ which are less stringent than those which I have formulated here. And it may be that we can satisfy these less stringent conditions without being able to satisfy those which I have formulated here.
An example for that is provided by the Born-Infeld electrodynamics, which is a modification of the Maxwell electrodynamics based on a different action integral, an action integral which is in agreement with the Maxwell one for weak fields, but differs from it for strong fields. This Born-Infeld electrodynamics leads to a classical K⊥ which involves square roots. It is of such a nature that it doesn’t seem possible to fulfill the conditions which are necessary for building up a relativistic quantum theory on curved surfaces. However, it does seem to be possible to build up a relativistic quantum theory on flat surfaces, for which the conditions are less stringent.
DR. DIRAC
Lecture No. 4
QUANTIZATION ON FLAT SURFACES
We have been working with states on general space-like curved surfaces in space-time. I will just summarize the results that we obtained concerning the conditions for a quantum field theory, formulated in terms of these states, to be relativistic. We introduce variables to describe the surface, consisting of the four coordinates YΛ of each point xr = (x1, x2, x3) on the surface. The x’s form a curvilinear system of coordinates on the surface. Then the y’s are treated as dynamical coordinates and there are momenta conjugate to them, wΛ(x), again functions of the x’s. And then we get a number of primary first-class constraints appearing in the Hamiltonian formalism, of the nature
(3-10)
The K’s are independent of the w’s, but may be functions of any of the other Hamiltonian variables. The K’s will involve the physical fields which are present. We analyze these constraints by resolving them into components tangential to the surface and normal to the surface. The tangential components are
(4-1)
and the normal components is
(4-2)
With this analysis, we find that the Kr can be worked out just from geometrical considerations. The Kr should be looked upon as something rather trivial, associated with transformations in which the coordinates of the surface are varied, but the surface itself doesn’t move. The first-class constraints (4-2) are associated with the motion of the surface normal to itself and are the important ones physically.
/> Certain Poisson bracket relations (3-21), (3-22), and (3-23) have to be fulfilled for consistency. Some of the Poisson bracket relations involve merely the Kr, and they are automatically satisfied when the Kr are chosen in accordance with the geometrical requirements. Some of the consistency conditions are linear in K⊥ and they are automatically satisfied provided we choose K⊥ to be a scalar density. Then finally we have the consistency conditions which are quadratic in the K⊥ and those are the important ones, the ones which cannot be satisfied by trivial arguments.
These important consistency conditions can be satisfied in the classical theory if we work from a Lorentz-invariant action principle and calculate the K⊥ by following the standard rules of passing from the action principle to the Hamiltonian. The problem of getting a relativistic quantum theory then reduces to the problem of suitably choosing the non-commuting factors which occur in the quantum K⊥ in such a way that the quantum consistency conditions are fulfilled, which means that the commutator of two of the quantities (4-2) at two points in space x1, x2, x3 has to be a linear combination of the constraints with coefficients occurring on the left. These quantum consistency conditions will usually be quite difficult to satisfy. It turns out that one can satisfy them with certain simple examples, but with more complicated examples it doesn’t seem to be possible to satisfy them. That leads to the conclusion that one cannot set up a quantum theory for these more general fields with the states defined on general curved surfaces.
I might mention that the quantities K have a simple physical meaning. Kr can be interpreted as the momentum density, K⊥ as the energy density; so the momentum density, expressed in terms of Hamiltonian variables, is something which is always easy to work out just from the geometrical nature of the problem and the energy density is the important quantity which one has to choose correctly (satisfying certain commutation relations) in order to satisfy the requirements of relativity.
If we cannot set up a quantum theory with states on general curved surfaces, it might still be possible to set it up with states defined only on flat surfaces.
We can get the corresponding classical theory simply by imposing conditions which make our previous curved surface into a flat surface. The conditions will be the following: The surface is specified by YΛ(x); in order to make the surface flat, we require that these functions shall be in the form
(4-3)
where the a’s and b’s are independent of the x’s. This will result in the surface being flat, and in the system of coordinates xr being rectilinear. At present we are not imposing the conditions that the xr coordinate system shall be orthogonal: I shall bring that in a little later. We are thus working with general, oblique, rectilinear axes xr.
We now have our surface fixed by quantities abΛr and these quantities will appear as the dynamical variables needed to fix the surface. We have far fewer of them than previously. In fact, we have only 4 + 12 = 16 variables here. We have these 16 dynamical coordinates to fix the surface instead of the previous YΛ(x), which meant 4·∞3 dynamical coordinates.
When we restrict the surface in this way, we may look upon the restriction as bringing a number of constraints into our Hamiltonian formalism, constraints which express the 4·∞3 y coordinates in terms of 16 coordinates. These constraints will be second-class. Their presence means a reduction in the number of effective degrees of freedom for the surface from 4·∞3 to 16, a very big reduction!
In a previous lecture I gave the general technique for dealing with second-class constraints. The reduction in the number of effective degrees of freedom leads to a new definition of Poisson brackets. This general technique is not needed in our present case, where conditions are sufficiently simple for one to be able to use a more direct method. In fact, we can work out directly what effective momentum variables remain in the theory when we have reduced the number of effective degrees of freedom for the surface.
With our dynamical coordinates restricted in this way, we have of course the velocities restricted by the equation
(4-4)
The dot refers to differentiation with respect to some parameter τ . As τ varies, this flat surface varies, moving parallel to itself and also changing its direction. The surface thus moves with a four-fold freedom, and this motion is expressed by our taking aΛ, bΛr to be functions of the parameter τ.
The total Hamiltonian is now
(4-5)
(I have taken the quantities , bΛ. outside the integral signs, because they are independent of the x variables.) (4-5) involves the ω variables only through the combinations ∫ wΛ d3 x and ∫ xr wΛd3x. We have here 16 combinations of the w’s, which will be the new momentum variables conjugate to the 16 variables a, b which are now needed to describe the surface.
We can again express HT in terms of the normal and tangential components of these quantities:
(4-6)
Let us now bring in the condition that the xr coordinate system is orthogonal. That means
(4-7)
Differentiating (4-7) with respect to τ, we get
(4-8)
(I have been raising the Λ suffixes quite freely because the Λ coordinate system is just the coordinate system of special relativity.) This equation tells us that is antisymmetric between r and s. So the last term in (4-6) is equal to
Now you see that we don’t have so many linear combinations of the w’s occurring in the HT as before. The only linear combinations of the w’s which survive are the following ones:
(4-9)
(4-10)
and also
(4-11)
and
(4-12)
(We can raise and lower the suffixes r quite freely now because they refer to rectilinear orthogonal axes.) These are the momentum variables which are conjugate to the variables needed to fix the surface when the surface is restricted to be a flat one referred to rectilinear orthogonal coordinates.
The whole set of momentum variables included in (4-9), (4-10), (4-11), and (4-12) can be written as Pµ and Mµν = − Mνµ, where the suffixes µ and ν take on 4 values, a value 0 associated with the normal component, and 1, 2, 3 associated with the three x’s. µ, ν are small suffixes referring to the x coordinate system, to distinguish them from the capital suffixes Λ referring to the fixed y coordinate system.
So now our momentum variables are reduced to just 10 in number, and associated with these 10 momentum variables we have 10 primary first-class constraints, which we may write
(4-13)
(4-14)
where
(4-15)
(4-16)
(4-17)
and
(4-18)
We have now 10 primary first-class constraints associated with a motion of the flat surface. In Lecture (3) I said that we would need 4 primary first-class constraints (3-10) to allow for the general motion of a flat surface. We see now that 4 is not really adequate. The 4 has to be increased to 10, because 4 elementary motions of the surface normal to itself and changing its direction would not form a group; in order to have these elementary motions forming a group, we have to extend the 4 to 10, the extra 6 members of the group including the translations and rotations of the surface, which motions affect merely the system of coordinates in the surface without affecting the surface as a whole. In this way we are brought to a Hamiltonian theory involving 10 primary first-class constraints.
We have now to discuss the consistency conditions, the conditions in terms of Poisson bracket relations which are necessary for all the constraints to be first-class. Let us first discuss the Poisson bracket relations between the momentum variables Pµ, Mµν. We are given these momentum variables in terms of the to variables (4-9) to (4-12), and we know the Poisson bracket relations (3-17), (3-18), and (3-19) between the w variables, so we can calculate the Poisson bracket relations between the P and M variables. It is not really necessary to go through all this work to determine the Poisson bracket relations between the P and M variables. It is sufficient to realize
that these variables just correspond to the operators of translation and rotation in four-dimensional flat space-time, and thus their Poisson bracket relations must just correspond to the commutation relations between the operators of translation and rotation. In either way we get the following Poisson bracket relations: