Second Quantization

The wave equations discussed in the last three chapters were able to describe the quantum mechanical behavior of single particles in a covariant manner. Such a treatment is referred to as first quantization. It is suitable for the description of the interactions of massive particles with kinetic energies much less than the particle rest mass, where energy conservation forbids the production of real particle-antiparticle pairs. However, at higher energies where the production of single particles (for cases when particle number is not conserved, such as for neutral pions, 7r°'s), or particle-antiparticle pairs (in cases where particle number is conserved) is energetically possible, the first quantized form fails completely, and we need to develop a new quantization scheme capable of describing particle production and annihilation fully. Such a quantization scheme is referred to as second quantization.

The quantum field theory of the EM field developed in Chapter 2 is just such a theory. In the case of the EM field, we started immediately with the second quantized (field) theory because a useful first quantized theory of photons does not exist. This is because photons have zero rest mass and photon number is not conserved, and therefore photons can always be created, no matter how small the energy. For classical particles the first quantized theory developed in the preceding three chapters was a useful development in itself and an essential first step to a more complete theory.

We are now ready to extend our previous treatment of the EM field to the description of classical particles, such as spin | electrons, quarks, or protons or spin zero pions. We proceed by first interpreting the single particle wave functions which emerge from the first quantized theory as "classical" fields and then turning these c-number fields into quantized q-number (operator) fields, just as we did for the photon. The resulting quantum fields have the same general structure as the EM quantum field, showing that classical particles and classical waves (photons) are ultimately described by the same mathematical object, a quantum field.

In this chapter we will discuss the construction of theories which describe free, non-interacting particles. A very important result will emerge. The requirement that the energy of free states be positive leads to the conclusion that the states of

spin \ particles must be antisymmetric and satisfy Fermi-Dirac statistics, while the states of spin 0 particles must be symmetric and satisfy Bose-Einstein statistics. This famous result is referred to as the connection between spin and statistics and is one of the great achievements of relativistic quantum field theory. We will conclude this chapter with a brief discussion of how interactions are included in quantum field theories. The study of interacting field theories will resume again in Chapter 9, after a discussion, in Chapter 8, of the role which symmetries play in the development of field theories.

7.1 SCHRODINGER THEORY

For comparison, we first discuss the second quantized form of the Schrodinger theory. The first step in this development is to regard the Schrodinger wave function, ip(x), as a classical field. A Lagrangian density which will yield a Schrodinger equation for this complex field is where the arrow over the operator shows in which direction it acts. If tp and ip* are independent (corresponding to two independent real fields), the Euler-Lagrange equations are where one of the two dip/dt terms comes from the derivative of £ with respect to dip*/dt and the other from the derivative of C with respect to ip*. Combining these terms gives the familiar Schrodinger equation for ip

The momentum conjugates to ip and ip* are'

Hence the Hamiltonian density is

Hence the Hamiltonian density is dip(x)

"In order to agree with the convention we will use later for Dirac fields, the momentum conjugate to i¡j will be denoted by n*, and not tt.

7.1 SCHRODINGER THEORY 189

and the total Hamiltonian, after integrating by parts and assuming the boundary terms vanish (because of the periodic boundary conditions we have always imposed; recall the discussion in Sec. 2.2), becomes

This is simply the expectation value of the kinetic energy operator, a result familiar from elementary studies.

Note that a popular alternative to the Lagrangian density (7.1) is

This Lagrangian density will also give the Schrodinger equation for ip but is not Hermitian and breaks the symmetry which naturally exists between i[i and ip*. In particular, it gives it* = iip* and n = 0 which is inconsistent with other relations. We will always use a Hermitian Lagrangian density.

For the free Schrodinger theory, the eigensolutions of the Schrodinger equation satisfy

and imposing the same periodic boundary conditions we used in Chapters 2 and 3, they are explicitly

where pn - L normalized, c,ny,nx) and = p£/(2m). These states are orthogonal and

We will now quantize this classical field theory. The general procedure, which was fully developed in Chapters 1 and 2, is to expand the field in terms of eigensolutions of the field equations and to interpret the expansion coefficients as annihilation and creation operators. For the Schrodinger theory, all > 0, so the most general expansion is

where ipi+\x) are positive energy normalized solutions (7.7) of the field equation (7.6) and an are annihilation operators with the following interpretation:

an destroys a particle of momentum pn a^ creates a particle of momentum pn .

These operators satisfy (for now) the commutation relations

[We shall see later that we could also use anticommutation relations with the Schrbdinger theory.]

Substituting the field expansion (7.9) into the expression (7.5) for H and using the Schrôdinger equation (7.6) and the orthogonality relations (7.8) give immediately

This form is familiar from Chapters 1 and 2, and all of the consequences we worked out in those chapters can be carried over to this case. The eigenstates of H are the Fock states, and (7.11) tells us that the total energy of any Fock state is simply the sum of the energies of each of the particles in that state. The equation tells us to first compute the number of particles with momentum pn (the number operator a^an), then multiply by the energy of a single particle with momentum pn, and finally add these contributions together.

The canonical commutation relations for this field theory are

which differs by a factor of | from what is expected (showing that it is better to use the [o, a*] commutation relations). To prove this, note that and

The last relation could be proved from the explicit form of the solutions, but it follows more generally from the completeness relation. Because of this factor of 2, the CCR's are usually written, for a theory of this type, in the form

Next, observe that the Hamiltonian has the required property of time translation, _

Next, observe that the Hamiltonian has the required property of time translation, _

The proof is simple and instructive. The commutator is

From the commutation relations it follows that

From the commutation relations it follows that

Hence

Note that the commutation relation (7.17) is a necessary and sufficient condition for the result (7.15). As long as the Hamiltonian has the form (7.11) and the relation (7.17) can be proved, the Hamiltonian will be the generator of time translations.

7.2 IDENTICAL PARTICLES

As we saw in Chapters 1 and 2 and in the preceding section, the quantization of a classical field leads immediately to creation and annihilation operators and to the introduction of Fock states which describe many particles. The particles associated with the quantization of a single field are identical. Quantum mechanically, this means that no measurement can be constructed which will distinguish them, and since the results of measurements in quantum theory are expressed as absolute squares of matrix elements, the requirement of indistinguishability takes the mathematical form where |lnil„2) is the Fock state of two identical particles, one with momentum rii and the other with momentum n2 (in general, we will use the notation \Nni) to denote a state of N particles with momentum ni), O is any operator, and (/| is any final state. Equation (7.19) is the statement that we can only know that one of the particles has momentum ni and the other has 712, but we cannot know which particle has which momentum (this is, in fact, a meaningless question). From Eq. (7.19) we conclude that where the phase factor must be ±1 if we assume that two interchanges necessarily carry us back to the same state. Since O and (/| are arbitrary, we obtain the result that the Fock states of a quantum field must be either symmetric or antisymmetric:

This in turn means that the creation operators (and therefore the annihilation operators as well) must satisfy either commutation relations or anticommutation relations. We have already discussed quantization with commutation relations and will now discuss anticommutation relations.

Anticommutation Relations

To construct a field theory based on either commutation or anticommutation relations, it is sufficient to require the following commutation relation:

Was this article helpful?

0 -1

Post a comment