Quantum Mechanics with Basic Field Theory

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Wilson in Wilson, He introduced a formulation of Quantum Chromodynamics on a space-time lattice, which allows the application of various non-perturbative techniques. This discretization will be explained in detail below. It leads to mathematically well-defined problems, which are at least in principle solvable. It should also be pointed out that the introduction of a space-time lattice can be taken as a starting point for a mathematically clean approach to quantum field theory, so-called constructive quantum field theory.

In modern quantum field theory, the introduction of a space-time lattice is part of an approach different from the operator formalism. This is lattice field theory. Its main ingredients are. Lattice field theory has turned out to be very successful for the non-perturbative calculation of physical quantities. In this Wiki an introduction and overview over the foundations and methods of lattice field theory is given.

The main concepts are here illustrated with a scalar field theory. The functional integral formulation of quantum field theory is a generalization of the quantum mechanical path integral. Perhaps this is the most intuitive picture of the quantum mechanical transition amplitude. It can be written as an integral over contributions from all possible paths from the starting point to the final point. Each path is weighted by the classical action evaluated along this path.

Unified field theory

For a detailed and mathematically rigorous account of path integrals the interested reader is referred to the textbook Glimm and Jaffe, The representation of quantum mechanics in terms of path integrals can be translated to field theory. In particular, S-matrix elements are related to Greens functions, e. Instead of discussing the functional integral representation for quantum field theory from the beginning, we shall restrict ourselves to translating the quantum mechanical concepts to field theory by means of analogy.

As mentioned before, any derivation of functional integrals is not attempted here, but just a motivation of their form by analogy. Furthermore, in the case of quantum mechanics the transition amplitude has been considered, whereas now the formula for Greens functions has been written, which is a bit different. The formulae for functional integrals give rise to some questions. Secondly, these integrals contain oscillating integrands, due to the imaginary exponents; what about their convergence?

Moreover, is there a way to evaluate them numerically? In the following it will be discussed, how the introduction of imaginary times helps in answering these questions. Return to quantum mechanics for a moment. Here one can also introduce Greens functions, e. In the following it will be demonstrated that these Greens functions are related to quantum mechanical amplitudes at imaginary times by analytic continuation.

This is the so-called Wick rotation, illustrated in Figure 3. Now we turn to field theory again. They are taken as starting point for non-perturbative investigations of field theories and for constructive studies. Whether it is possible to continue a specific field theory analytically from real to imaginary times and vice versa, depends on certain conditions to be satisfied.

For a large class of field theories these conditions have been analyzed and formulated by Osterwalder and Schrader, see Osterwalder and Schrader, , In particular, a Euclidean field theory must satisfy the so-called reflection positivity in order to correspond to a proper field theory in Minkowski space.

Prerequisite Knowledge

This makes Euclidean functional integrals so attractive compared to their Minkowskian counterparts. One might think that in the Euclidean domain everything is unphysical and there is no possibility to get physical results directly from the Euclidean Greens functions.

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But this is not the case. For example, the spectrum of the theory can be obtained in the following way. One central question still remains: does the infinite dimensional integration over all classical field configurations, i. In quantum mechanics the path integral representation can be derived as a limit of a discretization in time. It should be noted that the lattice spacing, being a dimensionful quantity, is not a parameter of the discretized theory, which could e. The size of the lattice spacing in physical units is a derived quantity determined by the dynamics.

This will be explained in Section "Continuum limit".

Learning outcomes

So a discrete set of variables has to be integrated. If the lattice is taken to be finite, one just has finite dimensional integrals. Relativity Made Relatively Easy.

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