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Fundamental theories of physics

Quantum Field Theory

Gerard ’t Hooft

Published Jan 1, 2016• 0 citations
Publisher

Abstract

Since the finiteness of the speed of light, and other considerations of special relativity, play such an important role in arguments concerning the interpretation of quantum mechanics, a good understanding of quantum field theory is a prerequisite. In this chapter, a brief summary is presented of what Quantum Field Theory is, and what one can conclude from it. The fundamental fields are bosonic fields, accounting for fundamental particles with spin 0 and 1, and fermionic fields describing fundamental particles with spin \(\frac{1}{2}\) . Within the framework of Quantum Field Theory, there are no conflicts between quantum mechanics and locality, causality, and positivity of the energy, but there seems to be no ‘underlying real world’. We do note the importance of correlations, even in the vacuum state. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-3-319-41285-6_20

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Since the finiteness of the speed of light, and other considerations of special relativity, play such an important role in arguments concerning the interpretation of quantum mechanics, a good understanding of quantum field theory is a prerequisite. In this chapter, a brief summary is presented of what Quantum Field Theory is, and what one can conclude from it. The fundamental fields are bosonic fields, accounting for fundamental particles with spin 0 and 1, and fermionic fields describing fundamental particles with spin (\frac{1}{2}) . Within the framework of Quantum Field Theory, there are no conflicts between quantum mechanics and locality, causality, and positivity of the energy, but there seems to be no ‘underlying real world’. We do note the importance of correlations, even in the vacuum state. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Quantum field theory is the ultimate expression of quantum mechanics. Its formalism provides the framework for all modern fundamental theories about the nature of matter. However, it involves difficult and intricate mathematics, which is why physics students don’t generally come across it until postgraduate level, and then only if their subject specialism requires the insights that it provides. In ordinary quantum theory, observable properties are represented by Hermitian operators, with eigenstates that form a basis for the expansion of a system’s states. Philosophers have been debating the correctness of such arguments, but now it seems that experimental science has come along and shown that, at the quantum level at least, the objects we study have no substance to them, independent of their properties. In classical physics, interactions between particles can be considered by either an examination of forces or from the potential energies involved.
The world of atoms and molecules is described by exactly the same expressions as the determinisric world described in the previous chapter, yet here, a world consisting of ‘real things’, seems to be missing. As soon as one makes some very basic assumptions about locality and causality, it seems to be possible to prove that such a world cannot exist at all. Have other assumptions been made? Can deterministic theories exist that generate strong correlations in space-like directions? What does free will mean? These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Fundamentals of quantum mechanics Quantum mechanics remains one of the most successful, yet enigmatic, formulations of physics. Uncertainty and measurement constraints are incorporated at the core of this fundamental description of micro-physical dynamics. Quantum mechanics successfully describes the observed structures in chemistry and materials science. In particular, the impenetrability of matter due to Pauli exclusion is a consequence of incorporating special relativity into quantum mechanics. Microscopic causality , or the requirement that communications cannot propagate at greater than the speed of light, relates a particle's spin to its quantum statistics but does not exclude the space-like correlations associated with a coherent quantum state. This section will examine some of the foundations of quantum physics. An emphasis will be placed upon how the microscopic fundamentals of quantum mechanics relate to the geometric fundamentals of gravitational mechanics. Quantum formalism There are several interpretations of quantum mechanics offering models of the underlying (often hidden) dynamics that generate the observed quantum phenomenology (for example, the Copenhagen interpretation, or the many-worlds interpretation). Since any interpretation consistent with quantum physics cannot be proved or disproved by experiment, only those aspects of quantum formalism directly connected to experimental observables will be developed, with minimal interpretation imposed. To establish the conventions utilized in what follows, a brief overview of the formalism that describes those calculable observables modeled by quantum physics will be given.
The non-linear Gross-Pitaevskii (GP) equation plays an important role in the mean-field theory of the Bose-Einstein condensation and has interesting mathematical properties. In this lecture we consider two nontrivial problems related to physical interpretation of the solutions of the GP equation: the superfluidity of a BEC gas in an optical lattice and the superfluidity of a 1D Bose-gas, with special attention to the role of grey solitons. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Abstract Much attention has been directed to the philosophical implications of quantum field theory (QFT) in recent years; this paper attempts a survey in low-technical terms. First the relations of QFT to other kinds of theory, classical and quantum, particle and field, are discussed. Then various formulations of QFT are introduced, along with related interpretations. Finally a review is made of some of the most interesting foundational problems. For our discussion of the foundations of quantum field theory (QFT) it will be useful to have a map showing where QFT stands in relation to other physical theories. Figure 1 locates theories as points of a three-dimensional lattice: describing particles or fields, formulated in different spacetimes and either classical or quantum. Consider the classical plane of Figure