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A map from $(q,p)$ to $(Q,P)$ is called an extended canonical transformation if it satisfies $$ \lambda(pdq-H(q,p,t)dt)-(PdQ-K(Q,P,t)dt)=dF $$ Here, to include the change of $t$, let us use $$ \lambda(pdq-H(q,p,t)dt)-(PdQ-K(Q,P,t')dt')=dF $$ instead.

Time reversal map is defined by $$ t'=-t,\quad Q(t')=q(t),\quad P(t')=-p(t) $$ This looks like an example of extended canonical transformation with $\lambda=-1$, $K(Q(t'),P(t'),t')=H(q(t),p(t),t)$, $F=0$. Am I correct?

Also, time translation is defined by $$ t'=t+\epsilon,\quad Q(t')=q(t),\quad P(t')=p(t) $$ This looks like an example of extended canonical transformation with $\lambda=+1$, $K(Q(t'),P(t'),t')=H(q(t),p(t),t)$, $F=0$. When people say time evolution is an example of canonical transformation (i.e., $\lambda=1$), is this what people usually mean?

Finally, can I discuss other transformation involving change of time, such as Lorentz transformation, in the same way?

I couldn't find these kind of transformations discussed in Hamiltonian formalism in textbooks. If anyone knows a good reference, please let me know.

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3 Answers 3

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  1. The notion of canonical transformations (CT) [which is traditionally defined in the context of a $2n$-dimensional symplectic manifold] can be defined in the setting of a $(2n+1)$-dimensional contact manifold where the time-coordinate is also transformed, cf. e.g. this Phys.SE post.

  2. However, the specific notion of extended CT [as it is defined in Ref. 1 and that OP seems to be referencing] does by definition not transform the time-coordinate.

References:

  1. H. Goldstein, Classical Mechanics; Chapter 9. See text under eq. (9.11).
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The solution of a canonical system is the problem, to find the generating function, that induces a canonical transformation from the start values $$(q(0),p(0)) \to (q(t),p(t)).$$ The generating function $F$, whose partial derivatives yield the canonical maps of coordinate and momentum variables, preserving Poisson brackets, is a solution of the Hamilton-Jacobi partial differential equation.

Any (classical e.g. Goldstein) textbook on Theoretical/Classical/Analytic Mechanics has a chapter about it, because its at the center of two other theories:

Classical Statistical Mechanics with its focus on flows of points in phase space.

Quantum Theory: in the Hamilton-Jacobi equation, squares of derivatives are replaced by second derivatives. Canonical transformations are replaced by unitary maps in Hilbert spaces.

This was central in the 'first quantization' principle in 1926 and was forgotten later by evolution of quantum theory into index gymnastics and greek letter spelling in the beginning, perturbation theory of QED and finally a branch of applied functional analysis, group representation theory and gauge theories of fiber bundles on manifolds.

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Not involving a transformation in time but extending instead a basis, means the situation in which one constructs the Hilbert space of states of a system with infinitely many degrees of freedom and, among them, cannonical, i.e. maximum, integrability. One of such situations arises in superstring theory where the Hilbert space of the (open) strings is a tensor product of the Hilbert spaces of different types of modes of different types of objects in the theory, such that an element of one of them is represented by an infinite number of of other elements. The reason this is possible is that in a case of the Hilbert space of superstring, all elements have to behave uunder Hermitian conjugation (equivalently being of positive or negative norm) and with hermitian products of all different combinations of elements being real. This forces an infinite number of the basis elements to lie in a single linear space of infinitely large dimension.

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