Editing Path integral formulation (section)

== Quantum action principle ==

In quantum mechanics, as in classical mechanics, the [[Hamiltonian (quantum mechanics)|Hamiltonian]] is the generator of time translations. This means that the state at a slightly later time differs from the state at the current time by the result of acting with the Hamiltonian operator (multiplied by the negative [[imaginary unit]], {{math|−''i''}}). For states with a definite energy, this is a statement of the [[de Broglie relation]] between frequency and energy, and the general relation is consistent with that plus the [[superposition principle]].

The Hamiltonian in classical mechanics is derived from a [[Lagrangian (field theory)|Lagrangian]], which is a more fundamental quantity in the context of [[special relativity]]. The Hamiltonian indicates how to march forward in time, but the time is different in different [[Frame of reference|reference frames]]. The Lagrangian is a [[Lorentz scalar]], while the Hamiltonian is the time component of a [[four-vector]]. So the Hamiltonian is different in different frames, and this type of symmetry is not apparent in the original formulation of quantum mechanics.

The Hamiltonian is a function of the position and momentum at one time, and it determines the position and momentum a little later. The Lagrangian is a function of the position now and the position a little later (or, equivalently for infinitesimal time separations, it is a function of the position and velocity). The relation between the two is by a [[Legendre transformation]], and the condition that determines the classical equations of motion (the [[Euler–Lagrange equation]]s) is that the [[action (physics)|action]] has an extremum.

In quantum mechanics, the Legendre transform is hard to interpret, because the motion is not over a definite trajectory. In classical mechanics, with [[discretization]] in time, the Legendre transform becomes
: <math> \varepsilon H = p(t)\big(q(t + \varepsilon) - q(t)\big) - \varepsilon L</math>
and
: <math> p = \frac{\partial L}{\partial \dot{q}},</math>
where the [[partial derivative]] with respect to <math>\dot q</math> holds {{math|''q''(''t'' + ''ε'')}} fixed. The inverse Legendre transform is
: <math> \varepsilon L = \varepsilon p \dot{q} - \varepsilon H,</math>
where
: <math> \dot q = \frac{\partial H}{\partial p},</math>
and the partial derivative now is with respect to {{mvar|p}} at fixed {{mvar|q}}.

In quantum mechanics, the state is a [[quantum superposition|superposition of different states]] with different values of {{mvar|q}}, or different values of {{mvar|p}}, and the quantities {{mvar|p}} and {{mvar|q}} can be interpreted as noncommuting operators. The operator {{mvar|p}} is only definite on states that are indefinite with respect to {{mvar|q}}. So consider two states separated in time and act with the operator corresponding to the Lagrangian:
: <math> e^{i\big[p \big(q(t + \varepsilon) - q(t)\big) - \varepsilon H(p, q) \big]}.</math>

If the multiplications implicit in this formula are reinterpreted as ''matrix'' multiplications, the first factor is
: <math> e^{-ip q(t)},</math>
and if this is also interpreted as a matrix multiplication, the sum over all states integrates over all {{math|''q''(''t'')}}, and so it takes the [[Fourier transform]] in {{math|''q''(''t'')}} to change basis to {{math|''p''(''t'')}}. That is the action on the Hilbert space – <em>change basis to {{mvar|p}} at time {{mvar|t}}</em>.

Next comes
: <math>e^{-i\varepsilon H(p,q)},</math>
or <em>evolve an infinitesimal time into the future</em>.

Finally, the last factor in this interpretation is
: <math>e^{i p q(t + \varepsilon)},</math>
which means <em>change basis back to {{mvar|q}} at a later time</em>.

This is not very different from just ordinary time evolution: the {{mvar|H}} factor contains all the dynamical information – it pushes the state forward in time. The first part and the last part are just Fourier transforms to change to a pure {{mvar|q}} basis from an intermediate {{mvar|p}} basis.

Another way of saying this is that since the Hamiltonian is naturally a function of {{mvar|p}} and {{mvar|q}}, exponentiating this quantity and changing basis from {{mvar|p}} to {{mvar|q}} at each step allows the matrix element of {{mvar|H}} to be expressed as a simple function along each path. This function is the quantum analog of the classical action. This observation is due to [[Paul Dirac]].<ref>{{harvnb|Dirac|1933}}</ref>

Dirac further noted that one could square the time-evolution operator in the {{mvar|S}} representation:
: <math> e^{i\varepsilon S},</math>
and this gives the time-evolution operator between time {{mvar|t}} and time {{math|''t'' + 2''ε''}}. While in the {{mvar|H}} representation the quantity that is being summed over the intermediate states is an obscure matrix element, in the {{mvar|S}} representation it is reinterpreted as a quantity associated to the path. In the limit that one takes a large power of this operator, one reconstructs the full quantum evolution between two states, the early one with a fixed value of {{math|''q''(0)}} and the later one with a fixed value of {{math|''q''(''t'')}}. The result is a sum over paths with a phase, which is the quantum action.