Download Quantum Field Theory II

Quantum Field Theory II
PHYS-P 622
Radovan Dermisek, Indiana University
Notes based on: M. Srednicki, Quantum Field Theory
Chapters: 13, 14, 16-21, 26-28, 51, 52, 61-68, 44, 53, 69-74, 30-32, 84-86, 75, 87-89, 29
1
Review of scalar field theory
Srednicki 5, 9, 10
2
The LSZ reduction formula
based on S-5
In order to describe scattering experiments we need to construct
appropriate initial and final states and calculate scattering amplitude.
Summary of free theory:
one particle state:
vacuum state is annihilated by all a’s:
then, one particle state has normalization:
normalization is Lorentz invariant!
see e.g. Peskin & Schroeder, p. 23
3
Let’s define a time-independent operator:
wave packet with width !
that creates a particle localized in the momentum space near
and localized in the position space near the origin.
(go back to position space by Furier transformation)
is a state that evolves with time (in the Schrödinger picture), wave
packet propagates and spreads out and so the particle is localized far from
the origin in at
.
for
in the past.
is a state describing two particles widely separated
In the interacting theory
is not time independent
4
A guess for a suitable initial state:
we can normalize the wave packets so that
Similarly, let’s consider a final state:
where again
and
The scattering amplitude is then:
5
A useful formula:
Integration by parts,
surface term = 0,
particle is localized,
(wave packet needed).
is 0 in free theory, but not in interacting one!
E.g.
6
Thus we have:
or its hermitian conjugate:
The scattering amplitude:
we put in time ordering
(without changing anything)
is then given as (generalized to n i- and n’ f-particles):
7
Lehmann-Symanzik-Zimmermann formula (LSZ)
Note, initial and final states now have delta-function normalization,
multiparticle generalization of
.
We expressed scattering amplitudes in terms of correlation
functions! Now we need to learn how to calculate correlation
functions in interacting quantum field theory.
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Comments:
we assumed that creation operators of free field theory would work
comparably in the interacting theory ...
acting on ground state:
is a Lorentz invariant number
we want
,
so that
is a single particle state
otherwise it would create a linear combination of the ground state and
a single particle state
we can always shift the field by a constant
so that
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one particle state:
is a Lorentz invariant number
we want
,
since this is what it is in free field theory,
correctly normalized one particle state.
creates a
we can always rescale (renormalize) the field by a constant
so that
.
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multiparticle states:
is a Lorentz invariant number
in general,
creates some multiparticle states. One can
show that the overlap between a one-particle wave packet and
a multiparticle wave packet goes to zero as time goes to
infinity.
see the discussion in Srednicki, p. 40-41
By waiting long enough we can make the multiparticle
contribution to the scattering amplitude as small as we want.
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Summary:
Scattering amplitudes can be expressed in terms of correlation
functions of fields of an interacting quantum field theory:
Lehmann-Symanzik-Zimmermann formula (LSZ)
provided that the fields obey:
these conditions might not be consistent with the original form of
lagrangian!
12
Consider for example:
After shifting and rescaling we will have instead:
13
Path integral for interacting field
based on S-9
Let’s consider an interacting “phi-cubed” QFT:
with fields satisfying:
we want to evaluate the path integral for this theory:
14
it can be also written as:
epsilon trick leads to additional factor;
to get the correct normalization we require:
and for the path integral of the free field theory we have found:
15
assumes
thus in the case of:
the perturbing lagrangian is:
counterterm lagrangian
in the limit
we expect
and
we will find
and
16
Let’s look at Z( J ) (ignoring counterterms for now).
Define:
exponentials defined by series expansion:
let’s look at a term with particular values of P (propagators) and V (vertices):
number of surviving sources, (after taking all derivatives) E (for external) is
E = 2P - 3V
3V derivatives can act on 2P sources in (2P)! / (2P-3V)! different ways
e.g. for V = 2, P = 3 there is 6! different terms
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V = 2, E = 0 ( P = 3 ):
!
! ! !
!! !
!
!
! ! ! ! ! !
3! 3! 2 2 2
!
dx1
!
1
i
" " " " " "
x1
2! 6 6 3! 2 2 2
1
=
24
!
1
i
x2
1
i
dx2 (iZg g)2 ∆(x1 − x2 ) ∆(x1 − x2 ) ∆(x1 − x2 )
symmetry factor
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V = 2, E = 0 ( P = 3 ):
!
! ! !
!! !
!
!
! ! ! ! ! !
3! 3! 3! 2
!
dx1
!
" " " " " "
x2
x1
2! 6 6 3! 2 2 2
1
=
8
!
1
i
1
i
1
i
dx2 (iZg g)2 ∆(x1 − x1 ) ∆(x1 − x2 ) ∆(x1 − x1 )
symmetry factor
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Feynman diagrams:
a line segment stands for a propagator
vertex joining three line segments stands for
a filled circle at one end of a line segment stands for a source
e.g. for V = 1, E = 1
What about those symmetry factors?
symmetry factors are related to symmetries of Feynman diagrams...
What about those symmetry factors?
20
Symmetry factors:
we can rearrange three derivatives
without changing diagram
we can rearrange three vertices
we can rearrange two sources
we can rearrange propagators
this in general results in overcounting of the number of terms that give the same
result; this happens when some rearrangement of derivatives gives the same match up
to sources as some rearrangement of sources; this is always connected to some
symmetry property of the diagram; factor by which we overcounted is the symmetry
factor
21
the endpoints of each propagator can be swapped
and the effect is duplicated by swapping the two vertices
propagators can be rearranged in 3! ways,
and all these rearrangements can be duplicated by
exchanging the derivatives at the vertices
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25
26
27
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All these diagrams are connected, but Z( J ) contains also diagrams
that are products of several connected diagrams:
e.g. for V = 4, E = 0 ( P = 6 ) in addition to connected diagrams we also
have :
and also:
and also:
30
All these diagrams are connected, but Z( J ) contains also diagrams
that are products of several connected diagrams:
e.g. for V = 4, E = 0 ( P = 6 ) in addition to connected diagrams we also
have :
A general diagram D can be written as:
additional symmetry factor
not already accounted for by symmetry
factors of connected diagrams; it is nontrivial
only if D contains identical C’s:
the number of given C in D
particular connected diagram
31
Now
is given by summing all diagrams D:
any D can be labeled by a set of n’s
thus we have found that
connected diagrams.
is given by the exponential of the sum of
imposing the normalization
means we can omit vacuum diagrams
(those with no sources), thus we have:
vacuum diagrams are omitted from the sum
32
If there were no counterterms we would be done:
in that case, the vacuum expectation value of the field is:
only diagrams with one source contribute:
(the source is “removed” by the derivative)
and we find:
we used
since we know
which is not zero, as required for the LSZ; so we need counterterm
33
Including
term in the interaction lagrangian results in a new type of
vertex on which a line segment ends
e.g.
corresponding Feynman rule is:
at the lowest order of g only
in order to satisfy
contributes:
we have to choose:
Note,
must be purely imaginary so that Y is real; and, in addition, the
integral over k is ultraviolet divergent.
34
to make sense out of it, we introduce an ultraviolet cutoff
and in order to keep Lorentz-transformation properties of the propagator
we make the replacement:
the integral is now convergent:
we will do this type of calculations later...
and indeed,
is purely imaginary.
after choosing Y so that
we can take the limit
Y becomes infinite
... we repeat the procedure at every order in g
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e.g. at
we have to sum up:
and add to Y whatever
term is needed to maintain
...
this way we can determine the value of Y order by order in powers of g.
Adjusting Y so that
means that the sum of all connected
diagrams with a single source is zero!
In addition, the same infinite set of diagrams with source replaced by ANY
subdiagram is zero as well.
Rule: ignore any diagram that,
when a single line is cut,
fall into two parts, one of which
has no sources. = tadpoles
36
all that is left with up to 4 sources and 4 vertices is:
37
finally, let’s take a look at the other two counterterms:
we get
we used integration by parts
it results in a new vertex at which two lines meet, the corresponding
vertex factor or the Feynman rule is
for every diagram with a propagator there is additional one with this vertex
Summary:
we have calculated
in
theory and expressed it as
where W is the sum of all connected diagrams with no tadpoles and at
least two sources!
38
Scattering amplitudes and the Feynman rules
based on S-10
We have found Z( J ) for the “phi-cubed” theory and now we can calculate vacuum
expectation values of the time ordered products of any number of fields.
Let’s define exact propagator:
short notation:
W contains diagrams with at least two sources
+ ...
thus we find:
39
4-point function:
we have dropped terms that contain
does not correspond to any
interaction; when plugged to LSZ,
no scattering happens
Let’s define connected correlation functions:
and plug these into LSZ formula.
40
at the lowest order in g only one diagram contributes:
S=8
derivatives remove sources in 4! possible ways, and label external legs in 3
distinct ways:
each diagram occurs 8 times, which nicely cancels the symmetry factor.
41
General result for tree diagrams (no closed loops): each diagram
with a distinct endpoint labeling has an overall symmetry factor 1.
Let’s finish the calculation of
y
z
putting together factors for all pieces of Feynman diagrams we get:
42
For two incoming and two outgoing particles the LSZ formula is:
and we have just written
terms of propagators.
in
The LSZ formula highly simplifies due to:
We find:
43
44
four-momentum is conserved in scattering process
Let’s define:
scattering matrix element
From this calculation we can deduce a set of rules for computing
.
45
Feynman rules to calculate
:
for each incoming and outgoing particle draw an external line and label it with
four-momentum and an arrow specifying the momentum flow
draw all topologically inequivalent diagrams
for internal lines draw arrows arbitrarily but label them with momenta so that
momentum is conserved in each vertex
assign factors:
1
for each external line
for each internal line with momentum k
for each vertex
sum over all the diagrams and get
46
Additional rules for diagrams with loops:
a diagram with L loops will have L internal momenta that are not fixed; integrate
over all these momenta with measure
divide by a symmetry factor
include diagrams with counterterm vertex that connects two propagators, each
with the same momentum k; the value of the vertex is
now we are going to use
to calculate cross section...
47
Lehmann-Källén form of the exact propagator
based on S-13
What can we learn about the exact propagator from general principles?
Let’s define the exact propagator:
The field is normalized so that
Normalization of a one particle state in d-dimensions:
The d-dimensional completeness statement:
identity operator in
one-particle subspace
Lorentz invariant phase-space differential
48
Let’s also define the exact propagator in the momentum space:
In free field theory we found:
it has an isolated pole at
with residue one!
What about the exact propagator in the interacting theory?
49
Let’s insert the complete set of energy eigenstates between the two fields;
for
we have:
ground state, 0 - energy
one particle states
multiparticle continuum of states
specified by the total three
momentum k and other
parameters: relative
momenta, ..., denoted
symbolically by n
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Let’s define the spectral density:
then we have:
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similarly:
and we can plug them to the formula for time-ordered product:
was your homework
we get:
or, in the momentum space:
Lehmann-Källén form of the exact propagator
it has an isolated pole at
with residue one!
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