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Quantum Action Principle

  • Julian Schwinger
Chapter

Abstract

Consider infinitesimal displacements of the time origin,
$$\bar t = t - \delta t,$$
(5.1.1)
and the implied unitary transformation,
$$U = 1 + \frac{i}{\hbar }Gt\;with\;Gt = - \delta tH$$
(5.1.2)
where H, the Hamiltonian operator, depends upon a set of variables for the system, v α(t), and possibly on t itself. When we shift the origin, the variables are redefined,
$${\upsilon _\alpha }(t) = {\bar \upsilon _\alpha }(\bar t)$$
(5.1.3)
or
$$\begin{array}{*{20}{c}} {{{{\bar{\upsilon }}}_{\alpha }}(t) = {{\upsilon }_{\alpha }}(t + \delta t) = {{\upsilon }_{\alpha }}(t) + \delta t\frac{d}{{dt}}{{\upsilon }_{\alpha }}(t)} \\ { = {{\upsilon }_{\alpha }}(t) - \delta {{\upsilon }_{\alpha }}(t),} \\ \end{array}$$
(5.1.4)
where
$$\delta {\upsilon _\alpha }(t) = \frac{1}{{i\hbar }}[{\upsilon _\alpha }(t),Gt]$$
(5.1.5)
so that
$$ - \delta t\frac{d}{{dt}}{\upsilon _\alpha }(t) = \frac{1}{{i\hbar }}[{\upsilon _\alpha }(t), - \delta tH]$$
(5.1.6)

Keywords

Wave Function Action Principle Hamilton Operator Schrodinger Equation Explicit Time Dependence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Julian Schwinger

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