36.5 Dynamics

36.5 Dynamics

Concept links · terms present in this machine draft; source roles are unverified: adjoint operator

To describe the time evolution of a quantum field theory system, it is generally easier to work with the Heisenberg picture (in which the time dependence is in the operators) than the Schr¨odinger picture (in which the time dependence is in the states). This is especially true in relativistic systems where one wants to as much as possible treat space and time on the same footing. It is however also true in the case of non-relativistic multi-particle systems due to the complexity of the description of the states (inherent since one is trying to describe arbitrary numbers of particles) versus the description of the operators, which are built simply out of the annihilation and creation operators.

In the Heisenberg picture the time evolution of an operator $\hat{f}$ is given by

$$\hat{f}(t)=e^{iHt}\hat{f}(0)e^{-iHt}$$

and such operators satisfy the diferential equation

$$\frac{d}{dt}\hat{f}=[\hat{f},-iH]$$

For the operators that create and annihilate states with momentum $p_j$ in the finite cutof formalism of section 36.2, $H=\hat{H}$ (given by equation 36.10) and we have

$$\frac{d}{dt}{a}^{†}(p_j,t)=[a^{†}(p_j,t),-i\hat{H}]=i\frac{p_j^2}{2m}{a}^{†}(p_j,t)$$

with solutions

$$a^{†}(p_j,t)=e^{i\frac{p_j^2}{2m}t}{a}^{†}(p_j,0)\text{\hspace{1em}(36.18)}$$

Recall that in classical Hamiltonian mechanics, the Hamiltonian function $h$ determines how an observable f evolves in time by the diferential equation

$$\frac{d}{dt}f=\{f,h\}$$

Quantization takes $f$ to an operator $\hat{f},$ and h to a self-adjoint operator $H$.

In our case the “classical” dynamical equation is meant to be the Schr¨odinger equation. In the finite cutof formalism, one can take as Hamiltonian

$$h=\sum _j\frac{p_j^2}{2m}A(p_j)\overline{A}(p_j)$$

Here the $A(p_j)$ should be interpreted as linear functions that on a solution given by $\alpha$ take the value ${\alpha }_j$ , and h a quadratic function on solutions that takes the value

$$h(\alpha )=\sum _j\frac{p_j^2}{2m}\alpha (p_j)\overline{\alpha }(p_j)$$

Hamilton’s equations are

$$\frac{d}{dt}A(p_j,t)=\{A(p_j,t),h\}=i\frac{p_j^2}{2m}A(p_j,t)$$

with solutions

$$A(p_j,t)=e^{i\frac{p_j^2}{2m}t}A(p_j,0)$$

In the continuum formalism, one can write

$$h={\int }_{-\infty }^{+\infty }\frac{p^2}{2m}A(p)\overline{A}(p)dp$$

which should be interpreted as a limit of the finite cutof version. We will not try and give a rigorous continuum interpretation of a quadratic product of distributions such as this one. As discussed at the end of section 36.3, the quantization of h can be given a rigorous interpretation as a bilinear form. We will however assume the $A(p_j,t)$ have as continuum limits distributions $A(p,t)$ that satisfy

$$\frac{d}{dt}A(p,t)=i\frac{p^2}{2m}A(p,t)$$

so have time-dependence

$$A(p,t)=e^{i\frac{p^2}{2m}t}A(p,0)$$

Note that this time-dependence is opposite to that of the Schr¨odinger solutions, since, as a distribution, $\frac{d}{dt}A(p,t)$ evaluated on a function $f$ is $A(p,t)$ evaluated on $\begin{array}{r}-\frac{d}{dt}f.\end{array}$

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