Moore3_useful eqns
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A reference sheet of useful formulae for free scalar quantum field theory. It covers Fourier transforms in 3 and 4 dimensions, Lorentz-invariant phase space, the Klein-Gordon Lagrangian and Euler-Lagrange equation, and the mode expansion with creation and annihilation operators. It also lists the Wightman, retarded and time-ordered (Feynman) two-point functions with their momentum-space forms. The file name suggests it relates to a Moore text or notes, but the authorship is not shown in the text.
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1 Useful Formulae, All in One Place
Fourier transformation in 3 dimensions:
φ(/vector x) =/integraldisplayd3/vector p
(2π)3e+i/vector p·/vector xφ(/vector p), (1)
φ(/vector p) =/integraldisplay
d3/vector x e−i/vector p·/vector xφ(/vector x). (2)
Performing integrals over space:
/integraldisplay
d3/vector x ei(/vector p+/vector q)·/vector x= (2π)3δ3(/vector p+/vector q), (3)
which forces /vector q=−/vector p. Similarly,
/integraldisplayd3/vector p
(2π)3e−i(/vector x+/vector y)·/vector p=δ3(/vector x+/vector y), (4)
forcing /vector x=−/vector y.
Fourier transformation in 4 dimensions: recall that
x·p≡ηµνxµpν=x0p0−/vector x·/vector p , (5)
so the signs in the exponents are switched:
φ(x) =/integraldisplayd4p
(2π)4e−ip·xφ(p), (6)
φ(p) =/integraldisplay
d4x eip·xφ(x). (7)
Lorentz invariant phase space for a particle of mass m:
/integraldisplayd4p
(2π)42π δ(p2−m2)θ(p0) =/integraldisplayd3/vector p
(2π)32E/vector p, E /vector p≡/radicalBig
/vector p2+m2. (8)
HereE/vector pis also written ω/vector porp0.
Free field theory:
S=/integraldisplay
d4xL/bracketleftBig
φ(x), ∂µφ(x)/bracketrightBig
, (9)
LKlein −Gordon =1
2∂µφ∂µφ−m2
2φ2, (10)
πµ≡δL/bracketleftBig
φ, ∂µφ/bracketrightBig
δ∂µφ→K−G∂µφ , (11)
Euler-Lagrange :δL/bracketleftBig
φ, ∂µφ/bracketrightBig
δφ=∂µπµ→K−G∂µ∂µφ=−m2φ . (12)
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The field can be expanded in creation an annihilation operato rs:
φ(x) =/integraldisplayd3/vector p
(2π)32E/vector p/bracketleftBig
a/vector pe−ip·x+a†
/vector pe+ip·x/bracketrightBig
, (13)
/bracketleftBig
a/vector p, a†
/vector q/bracketrightBig
= 2E/vector p(2π)3δ3(/vector p−/vector q). (14)
Note that xand the phases here are 4-vector and 4-vector product, with p0≡E/vector p=√/vector p2+m2.
Two-point correlation functions (the book uses Drather than G):
G>(x−y)≡ /angbracketleft0|φ(x)φ(y)|0/angbracketright, (15)
G<(x−y)≡ /angbracketleft0|φ(y)φ(x)|0/angbracketright, (16)
GR(x−y)≡ /angbracketleft0|/bracketleftBig
φ(x), φ(y)/bracketrightBig
|0/angbracketrightΘ(x0−y0)
= (G>(x−y)−G<(x−y))Θ(x0−y0), (17)
GT(x−y)≡ /angbracketleft0|φ(x)φ(y)|0/angbracketrightΘ(x0−y0) +/angbracketleft0|φ(y)φ(x)|0/angbracketrightΘ(y0−x0)
=G>(x−y)Θ(x0−y0) +G<(x−y)Θ(y0−x0)
=/angbracketleft0|T/parenleftBig
φ(x)φ(y)/parenrightBig
|0/angbracketright. (18)
Free theory values:
G(x−y) =/integraldisplayd4p
(2π)4e−ip·xG(p) (19)
with
G>(p) = 2 πδ(p2−m2)Θ(p0) (20)
G<(p) = 2 πδ(p2−m2)Θ(−p0) (21)
GR(p) =i
p2−m2+iǫsign(p0)(22)
GT(p) =i
p2−m2+iǫ. (23)
GTis also written GF,DF, ∆Fetc and is called the time ordered correlator or the Feynman
correlator.
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