Talk:Ordered exponential

Sketch of derivation
It would be nice to see a sketch of the derivation, that each method of definition is equivalent to the same thing, and where ordering becomes relevant. Cesiumfrog (talk) 01:52, 14 March 2010 (UTC)

Confusion
I think there is some confusion in the definitions: in the first one t1<t2, in the last one t2<t1... (12:52, 25 January 2011 UTC) —Preceding unsigned comment added by 192.33.102.156 (talk)


 * It seems that this was removed. But it was correct, because during the ordering, you have to consider both possibilities: that t1<t2 and also that t2<t1 and then realize the expression is symmetric, and thus insert a factor of 1/2 to avoid double-counting, or 1/n! for the general case. The article on Dyson series explains this more clearly.

Ambiguous notation
In the definition formula after "The ordered exponential of a is denoted", it is unclear what each of the upper limits of integrals on the right:  t, ?, ?, t'n Chris2crawford (talk) 12:58, 31 October 2015 (UTC)
 * Fixed, thanks for pointing this out Chris2crawford Quantumavik (talk) 14:40, 8 July 2018 (UTC)

Commutation relations?
This article, as currently written, does not talk about commutation relations. And this is a good thing, sort of. Only sort-of, because this leaves the situation not entirely clear. The article on path-ordering explicitly states that operators commute either as bosons or as fermions, so that (I quote from that article):


 * For two operators A(x) and B(y) that depend on spacetime locations x and y we define:
 * $$\mathcal T \left\{A(x) B(y)\right\} := \begin{cases} A(x) B(y) & \text{if } \tau_x > \tau_y, \\ \pm B(y)A(x) & \text{if } \tau_x < \tau_y. \end{cases} $$
 * Here $$\tau_x$$ and $$\tau_y$$ denote the invariant scalar time-coordinates of the points x and y.

This is referenced with Weinberg, and is absolutely correct for conventional physics super-symmetric-style operators. For this conventional case, one only needs to track a parity sign $$\epsilon=\pm1$$ and everything is good.

This article, as currently written, appears to not require the above assumption (which is a good thing): it seems to allow for the time-reversed $$\tau_x < \tau_y$$ case, that $$\mathcal T \left\{A(x) B(y)\right\} = \pm B(y)A(x) + C(x,y)$$ for either some c-number $$C(x,y)\ne 0$$, or perhaps some generic algebra elt $$C(x,y)$$. The only problem here is that, since this is not made entirely explicit, its not clear if there are some dragons here, or not. Not clear if some implicit assumptions are being made.

See operator product expansion for a stubby sketch of the general formulation for non-trivial commutation rules. 67.198.37.16 (talk) 22:50, 26 May 2024 (UTC)