Quantization in the functional theory of particles

Radi A. Jishi

Nothing can be made out of nothing. William Shakespeare , King Lear Historically, quantization of the motion of particles was developed first. The state was described by a wave function and observables by operators. When dealing with interactions between particles and fields, such as the electromagnetic field, the fields were treated classically. Classical field equations look like the quantum mechanical equations for the wave function of the field quanta. For example, the Klein–Gordon classical field equation is similar to the quantum mechanical wave equation for a relativistic spinless particle. Quantizing the fields, leading to quantum field theory, appears to be quantizing a theory that has already been quantized; hence the name “second quantization.” In reality, there is only one quantization and one quantum theory. The method of second quantization is important in the study of many-particle systems. It enables us to express many-body operators in terms of creation and annihilation operators, thus rendering calculations less cumbersome. Moreover, the method makes it possible to treat systems with a variable number of particles; that is why the method initially emerged in the context of quantum field theory. In Chapter 1 we indicated that any one-particle wave function may be expanded in a complete set of states. In this chapter, we show that products of single-particle states, when properly symmetrized, form an orthonormal basis for the expansion of the wave function of an N -particle system.

Michele Cini

H. D. Zeh

The appearance of single-particle wave functions as field operators or arguments of a field functional in quantum field theory is often regarded as a second quantization, since these wave functions were themselves obtained by quantizing mass points (“particles”). I argue here that this language, though reflecting the historical development, is misleading. Spatial fields must be regarded as the true and only physical variables (in quantum theory becoming the arguments of a fundamental wave functional). This includes spinor and other fields which never become classical. Apparent particles are no more than the result of decoherence in the measuring device or by means of other environmental scattering processes, without playing any fundamental conceptual rôle in the theory or its interpretation. A remark on gauge fields is added.

R. MacKenzie, Prasanta Kumar Panigrahi, Manu B. Paranjape +1 more

Hiroyasu Koshimizu

Peng Dai

String theory uses the first-quantized method (quantum mechanics) to derive scattering amplitudes. External states are considered as vertex operators inserted on the worldsheet generated by an internal string and the amplitudes are calculated perturbatively by calculating vacuum expectation values (vevs) of these vertex operators on worldsheets with different topologies. This approach is different from the common approach of particle theories in which the second-quantized method quantum field theory) is adopted to calculate amplitudes. A natural question to ask is whether there exists a first-quantization formalism for particles that gives particle amplitudes to all orders. This thesis presents the recent research in answering this question. In the first-quantized approach for particles, amplitudes are considered as the vevs of vertex operators inserted on different graphs generated by an internal particle. I refer to these graphs as world-graphs and first-quantized approach for particles as worldgraph approach. To evaluate these vevs, vertex operators for several external states and Green functions on different worldgraphs are needed. In this thesis, various vertex operators are considered and a general method to obtain scalar Green functions on different worldgraphs is obtained. Some examples of the worldgraph approach to amplitudes in scalar theory and Yang-Mills theory are presented.