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Quantum Field Theory
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The approach initiated by Umezawa is embedded in the framework of quantum field theory, more broadly applicable and formally more sophisticated than standard quantum mechanics. It refers directly to the activity of neuronal assemblies as the neural correlates of mental representations. A clear conceptual distinction between brain states and mental states is most often missing, although the approach is not intended to be reductionistic. Vitiello's more recent accounts offer some clarifying hints in that direction, which point to an understanding in terms of a dual-aspect approach. Other such approaches, like those of Pauli and Jung and of Bohm and Hiley, are conceptually more transparent in this respect. On the other hand, they are essentially unsatisfactory with regard to a sound formal basis and concrete empirical scenarios.
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The quantum field theory approach visualizes the force between the electrons as an exchange force arising from the exchange of virtual photons. It is represented by a series of Feynman diagrams, the most basic of which is
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Without a solid understanding of quantum field theory, no student can claim to have mastered contemporary theoretical physics. Offering a remarkably accessible conceptual introduction, this text will be widely welcomed and used.
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In very loose terms, the operator valuedness of quantum fields means that to each space-time point (x,t) a field value φ(x,t) is assigned which is an operator. This is the fundamental difference to classical fields because an operator valued quantum field φ(x,t) does not by itself correspond to definite values of a physical quantity like the strength of the electromagnetic field. On this background, Teller has argued in Teller 1995 that the field interpretation of QFT is inappropriate since the alleged fields in QFT are not to be interpreted as physical fields with definite values of some sort which are assigned to space-time points, like in the case of the classical electromagnetic field. Rather, quantum fields are what Teller calls ‘determinables’ (p. 95), as it becomes manifest by the fact that quantum fields are described by mappings from space-time points to operators. Operators are mathematical entities which are defined by how they act on something.
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Besides general relativity and quantum field theory as usually practiced, a third sort of idealization of the physical world has attracted a great deal of attention in the last decade. These are called topological quantum field theories, or `TQFTs'. In the terminology of the previous section, a TQFT is a background-free quantum theory with no local degrees of freedom1.
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Up to this point, the aim was to develop a free field theory. Doing so does not only neglect interaction with other particles (fields), it is even unrealistic for one free particle because it interacts with the field that it generates itself. For the description of interactions—such as scattering in particle colliders—we need certain extensions and modifications of the formalism as so far exposed. The immediate contact between scattering experiments and QFT is given by the scattering or S-matrix which contains all the relevant predictive information about, e.g., scattering cross sections. In order to calculate the S-matrix the interaction Hamiltonian is needed. The Hamiltonian can in turn be derived from the Lagrangian density by means of the so-called Legendre transformation.
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Quantum Field Theory