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title=Chiral Perturbation Theory to One Loop |
journal=[[Annals of Physics]] | volume= 158 |issue=1 | pages= 142–210 | year=1984 |
doi= 10.1016/0003-4916(84)90242-2 |bibcode = 1984AnPhy.158..142G |url=https://cds.cern.ch/record/147992 |url-access=subscription }}</ref><ref name="condensates">{{ cite journal |
author=S. D. Glazek |
title=Light Front QCD in the Vacuum Background |
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where the relations are taken at equal time <math> t </math>, and <math> \vec x </math> and <math> \vec y </math> are the space variables. The equal-time requirement imposes that <math> \vec x - \vec y </math> is a [[Spacetime#Spacetime_interval|spacelike]] quantity. The non-zero value of the commutator <math>[\phi(t, \vec x),\pi(t, \vec y)]</math> expresses the fact that when <math> \phi </math> and <math> \pi </math> are separated by a spacelike distance, they cannot communicate with each other and thus commute, except when their separation <math> \vec x - \vec y \to 0</math>.<ref>{{cite book | last=Carroll | first=Sean | title=Spacetime and Geometry: An Introduction to General Relativity | publisher=Addison Wesley | year=2003 | isbn=0-8053-8732-3 | edition=Reprinted 2019 }}</ref>
In the Light-Front form however, fields at equal time <math> x^+ </math> are causally linked (i.e., they can communicate) since the Light-Front time <math> x^+ \equiv t-z </math> is along the light
<math display="block">{\rm Light-Front~form:}~~[\phi(x^+, \vec x),\phi(x^+, \vec y)] = \frac{i}{4}\epsilon(x^- -y^-)\delta^2( \vec{x_\bot} - \vec{y_\bot}),</math>
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