Canonical transformations in QFT (lecture notes) by Blasone M.

By Blasone M.

During this lecture notes, we talk about canonical alterations within the context of Quantum box idea (QFT).The objective isn't really that of supply a whole and exhaustive therapy of canonical ameliorations from a mathematical standpoint. quite, we are going to try and convey, via a few concrete examples, the actual relevance of those differences within the framework of QFT. This relevance is on degrees: a proper one, during which canonical alterations are a major software for the certainty of easy facets of QFT, corresponding to the life of inequivalent representations of the canonical commutation family members (see x1.2) or the way symmetry breaking happens, via a (homogeneous or non-homogeneous) condensation mechanism (see part 4), however, also they are valuable within the examine of particular actual difficulties, just like the superconductivity (see x2.2) or the sphere blending (see part 5).

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26) k From this last equation we see how the time evolution can be expressed in terms of only one subsystem, which is then regarded as an ”open” one. The formal interpretation of S as an entropy finds its justification in the following relations 0(t)|S|0(t) = − Wn (t) ln Wn (t) . 27) n≥0 where the Wn (t) are some coefficients [5] (see also Section 2). 28) which shows that 2i h ¯ ∂S is the generator of time-translations. The fact that the entropy operator S ∂t controls the time evolution is a signal of the irreversibility of such an evolution for the dissipative system under consideration.

This analysis will let us understand better the difference between QM and systems with an infinite number of degrees of freedom. We can define an invariance transformation for the dynamics as an automorphism of the algebra A of the canonical variables (Heisenberg algebra). ) and in particular the commutation relations, then it is a canonical transformation: the equations of motion, which in the Heisenberg representation are algebraic relation among the elements of A, are invariant under its action.

43) Beyond the tree level, in general the form of this equation changes: quantum corrections can affect the macroscopic level in a relevant way. 42), which becomes a series in f (x) only, since all the terms containing ϕ(x) are normal ordered, thus having zero expectation value on |0 . e. 43). 39) describes indeed, at higher orders, the dynamics of one or more quantum physical particles in the potential generated by the macroscopic object φf (x). 40), by expanding around the classical field φ(x) (including in general quantum corrections): δ 1 φf (x) + : δf (y) 2 1 (2) (1) ≡ φf (x) + ψf (x) + : ψf (x) : + ...

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