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Quantum Field Theory: Particles
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The third thread in the development of quantum field theory was the need to handle the statistics of many-particle systems consistently and with ease. In 1927, Jordan tried to extend the canonical quantization of fields to the many-body wavefunctions of identical particles, a procedure that is sometimes called second quantization. In 1928, Jordan and Eugene Wigner found that the quantum field describing electrons, or other fermions, had to be expanded using anti-commuting creation and annihilation operators due to the Pauli exclusion principle. This thread of development was incorporated into many-body theory, and strongly influenced condensed matter physics and nuclear physics.
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Quantum field theory solves these problems by consistently quantizing a field. By interpreting the physical observables of the field appropriately, one can create a (rather successful) theory of many particles. Here is how it is:
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"Over the years I have used parts of Srednicki's book to teach field theory to physics graduate students not specializing in particle physics. This is a vast subject, with many outstanding textbooks. Among these, Srednicki's stands out for its pedagogy. The subject is built logically, rather than historically. The exposition walks the line between getting the idea across and not shying away from a serious calculation. Path integrals enter early, and renormalization theory is pursued from the very start...By the end of the course the student should understand both beta functions and the Standard Model, and be able to carry through a calculation when a perturbative calculation is called for."
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In the 1960s, Ricciardi and Umezawa (1967) suggested to utilize the formalism of quantum field theory to describe brain states, with particular emphasis on memory. The basic idea is to conceive of memory states in terms of states of many-particle systems, as inequivalent representations of vacuum states of quantum fields.[10] This proposal has gone through several refinements (e.g., Stuart et al. 1978, 1979; Jibu and Yasue 1995). Major recent progress has been achieved by including effects of dissipation, chaos, and quantum noise (Vitiello 1995; Pessa and Vitiello 2003). For readable nontechnical accounts of the approach in its present form see Vitiello (2001, 2002).
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In quantum field theories, Hamiltonians are written in terms of either the creation and annihilation operators or, equivalently, the field operators. The former practice is more common in condensed matter physics, whereas the latter is more common in particle physics since it makes it easier to deal with relativity. An example of a Hamiltonian written in terms of creation and annihilation operators is
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In quantum field theory, the energy is given by the Hamiltonian operator, which can be constructed from the quantum fields; it is the generator of infinitesimal time translations. (Being able to construct the generator of infinitesimal time translations out of quantum fields means many unphysical theories are ruled out, which is a good thing.)In order for the theory to be sensible, the Hamiltonian must be bounded from below. The lowest energy eigenstate (which may or may not be degenerate) is called the vacuum in particle physics and the ground state in condensed matter physics (QFT appears in the continuum limit of condensed matter systems).
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