January, 2005
Abstract
It is generally known that for the construction of quantized fields it appears to be useful to calculate Poisson brackets of field values. And a Poisson bracket is a notion of the Hamiltonian formalism.
If we have a problem to create a completely relativistic-invariant scheme of quantization of fields, then we wish, first of all, to formulate the Hamiltonian formalism in a relativistic-invariant form.
Science solved this problem for surprisingly long time. In fact, it was generally accepted that the Hamiltonian formalism can not be formulated in an explicitly relativistic-invariant form.
Nevertheless, with development of methods of the symplectic geometry, it became possible to formulate Hamiltonian formalism on the basis of such notions that turned out to have relativistic-invariant analogs. Here these notions are: a phase space, a symplectic structure on the phase space, a canonical action of the one-parameter group of time shifts in the phase space.
For relativistic fields the analogs of these notions are, correspondingly: an invariant phase space, a symplectic structure on the invariant phase space, a canonical action of the Poincaré group on the invariant phase space.
A point of the usual phase space describes the dynamical state of a system in a fixed moment of time. If for each initial state of the system the equations of motion are solvable, and uniquely, then we can speak about one-to-one correspondence between the phase space of the system (for a fixed moment of time) and the set of solutions of the equations of motion. This set of solutions of the equations of motion is called an invariant phase space.
The invariant phase space possesses a natural structure of manifold. If the dynamical system is a relativistic field, then as possible coordinate functions on invariant phase space we can use values of the magnitude of the field in fixed points of space-time.
So, from the point of view of the structure of the invariant phase space, possible values of the field in a fixed point of space-time are just one of the huge set of functions on the invariant phase space.
It can seem that the absence of any special coordinate functions on the invariant phase space, that could be used to identify points of this space, makes this space an empty abstraction.
But this is not so. Because it is possible to show that the invariant phase space, like the usual one, possesses a symplectic structure.
When we state the one-to-one correspondence of points of the invariant phase space and points of a usual phase space, taken in a fixed moment of time, it turns out that their symplectic structures correspond to each other. From this it is clear that the existence of the symplectic structure on the invariant phase space is a fact which is a generalization of the Liouville and the Poincaré theorems.
As regards Poisson brackets, they are defined through the symplectic structure. Their mathematical definition, in fact, does not change. But so far as this definition is applied to objects of other nature, it turns out that such Poisson brackets are a deep generalization of the usual. For example, they can be calculated between values of the field in different moments of time.
If the initial Lagrangian of the field is relativistic-invariant, then the equations of motion are also relativistic-invariant. So, under any transformation from the Poincaré group a solution of the equations of motion is transformed into a solution.
In other words, the Poincaré group acts on the invariant phase space.
So far as the symplectic structure on the invariant phase space is defined through the Lagrangian, it turns out to be invariant under the action of the Poincaré group. So, the Poincaré group acts on the invariant phase space canonically.
For the problem of quantization we can restrict ourself with linear fields. These fields are characterized by the property that linear combination of solutions of equations of motion is a solution. From the point of view of the invariant phase space, we can say that in this case it possesses a natural linear structure.
The Poincaré group preserves this linear structure. So, the Poincaré group acts on the invariant phase space as a group of symplectic transformations.
So far as for the purpose of quantization it is necessary to analyse this action of the Poincaré group by methods of the group theory, it is useful to use the following terminology. We say that linear relativistic fields define symplectic representations of the Poincaré group. These representations (and also their conjugate and complexified) are also called field representations.
We will not become more profound here in the theory of field representations. Let us just notice that this theory is quite analogous to the Wigner-Mackey theory of unitary representations of the Poincaré group, but it plays a more fundamental role for the theory of quantized fields.
So, now we already have all necessary structures that are used for construction of the quantized field. Here they are: the invariant phase space of the linear classical field, the symplectic structure on that space, the properly classified invariant (with respect to the Poincaré group) subspaces of the field representation, and the proper set of classical field values that are “quantized”.
In this review I omit a description of the exact construction of a quantized field. From the point of view of an algebraist, all methods used for it are well known. Similar methods are used for construction of universal enveloping algebras for Lie algebras, for construction of Grassmanian algebras etc.
Let us give now a list of some most important properties of the quantization under consideration.
The problem of quantization of the electromagnetic field was one of the main stimuli for the creation of the RCQ.
It is known for a long time, that for the problems of the quantum theory it is necessary to describe the electromagnetic field by the means of the vector potential. It is also known that an attempt of quantization of such a vector field leads to the necessity of consideration of an indefinite scalar product in the quantum space of states.
An indefinite scalar product, in contrast to the positive-definite case, does not define a topology.
This problem was just ignored up to now (in the educational literature). By the analogy with some other fields, it was believed that the space of states of the quantized electromagnetic field must be a Hilbert space, at least from the topological point of view.
The RCQ method has shown that this is not so.
The RCQ method, of course, can be applied to other fields (for example, to the scalar, to the electron-positron etc.)
It is known that, for example, the scalar field has only one quantization in the Hilbert space. Such a quantization is already constructed, of course. And the RCQ method (if we restrict ourself with a positive-definite scalar product) can not lead to other quantization.
Of course, the RCQ method leads in this case to an equivalent quantization. But the undoubted merit of the method is that the construction is performed in the frame of the general scheme, without any “guesses”, “classical analogies” etc.
It happens so, because the RCQ method is a rigorous mathematical construction.