Path order

Path ordering is a mathematical operation commonly used in theoretical physics , characterized by the path ordering operator . Path order allows the generalization of certain series developments to non- commutative algebraic structures as they occur in quantum theory and quantum field theory . Roughly speaking, the lack of interchangeability of the operators in products creates a natural “order” that can be expressed in a compact way using path order. ${\displaystyle {\mathcal {P}}}$

In non- relativistic theories, in particular, time order , i.e. H. Path order according to the parameter time, of importance. This is indicated by the time order operator or . ${\displaystyle {\mathcal {T}}}$ ${\displaystyle T}$

The path order operator (and thus also the time order operator ) is not a linear operator and is therefore sometimes referred to as a “meta operator” or “symbol”.

definition

For a product of linear operators that depend on a parameter , the path-ordered product is defined as that permutation of the factors ${\displaystyle O_{i}(x_{i})}$ ${\displaystyle x}$ ${\displaystyle \pi }$

${\displaystyle {\mathcal {P}}O_{1}O_{2}\cdots O_{n}:=(\pm 1)^{\pi }O_{\pi (1)}O_{\pi (2)}\cdots O_{\pi (n)}}$,

so that the operators appear in order of the value of the parameter:

${\displaystyle x_{\pi (1)}>x_{\pi (2)}>\cdots >x_{\pi (n)}}$

If a parameter value occurs more than once, the path order is not defined. However, since path-ordered products are usually integrated via the parameter, the size of such points disappears . The sign for bosons is always +1, for fermions it is the same as the sign of the permutation (+1 if the number of exchanges is even, otherwise −1).

Example: Causal Green's function

In theoretical solid state physics , the causal Green function is important, which indicates the propagation of an electron or a hole in time. In the second quantization , this function can be written in compact form using the time order: ${\displaystyle G(t-t^{\prime })}$ ${\displaystyle (t>t^{\prime })}$ ${\displaystyle (t<t^{\prime })}$

${\displaystyle G(t-t^{\prime })=-\mathrm {i} \left\langle {\mathcal {T}}\;\psi (t)\,\psi ^{\dagger }\!(t^{\prime })\right\rangle ={\begin{cases}-\mathrm {i} \left\langle \psi (t)\,\psi ^{\dagger }\!(t^{\prime })\right\rangle &t>t^{\prime }\\+\mathrm {i} \left\langle \psi ^{\dagger }\!(t^{\prime })\,\psi (t)\right\rangle &t<t^{\prime }\\\end{cases}}}$

Path-ordered exponential

Time order often occurs within a series expansion. The time-ordered exponential function has become established here:

${\displaystyle {\begin{aligned}{\mathcal {T}}\exp \left(\int _{0}^{t}d\tau A(\tau )\right):&=\sum _{n=0}^{\infty }{\frac {1}{n!}}\iint \!\cdots \!\int _{0}^{t}d^{n}\tau \ {\mathcal {T}}\left[A(\tau _{1})A(\tau _{2})\cdots A(\tau _{n})\right]\\&=\sum _{n=0}^{\infty }\int _{0}^{t}d\tau _{1}\int _{0}^{\tau _{1}}d\tau _{2}\cdots \int _{0}^{\tau _{n-1}}d\tau _{n}\ A(\tau _{1})A(\tau _{2})\cdots A(\tau _{n})\end{aligned}}}$

This can be generalized to any functions of the operator.

credentials

• Alexandre M. Zagoskin: Quantum Theory of Many-Body Systems . In: Graduate Texts in Contemporary Physics . Springer, New York, NY 1998, ISBN 978-1-4612-6831-4 , pp. 24 .