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Feynman Path Integrals

Feynman Path Integrals are an alternative formulation of quantum field
theory. Unlike the canonical approach which is based on creation and
annihilation operators, the path integral approach generalizes the
action principle of classical mechanics. The use of path integrals
replaces the classical notion of a single, unique history for a system
with a sum over an infinity of possible set of paths (histories) to
compute a quantum amplitude. Feynman showed that an amplitude computed
in this way will also obey the Schrodinger equation for the Hamiltonian
corresponding to the given action.
Dirac pointed out that the propagator was analogous to the quantity
exp(iS/h) where S is the time integral of the Lagrangian in classical
mechanics. Using this, Feynman postulated that the probability
amplitude for a particular spacetime path is given by:
exp(iS_{Γ}/h)
Where S_{Γ} = ∫Ldt, the classical action calculated along a path, Γ.
The propagator is given by:
K(x,y) = ∫exp(iS_{Γ}/h)Dx
= ∫exp((i/h)∫Ldt)Dx
Where Dx represents all possible paths in spacetime.
In order to find the overall probability amplitude for a given process it
is necessary to sum over the space of all possible spacetime paths of
the system in between the initial and final states, including paths that
are absurd by classical standards. This different to the classical case
where the goal is to minimize the action along a single trajectory.
The initial and final states are then related to each other by:
φ(x) = ∫dy K(x,y)φ(y)
Green's Functions

Consider the differential equation:
Du(x) = f(x)
Where D is a linear differential operator.
The Green's function, G(x,u) of D is defined by the equation:
DG(x,u) = δ(x  u)
Where u is a dummy variable used to describe the position of the δ
function.
The solution for u(x) is given by:
_{∞}
u(x) = ∫duG(x,u)f(u)
^{0}
This is the same form as the Feynman propagator, K.
Time Slicing

The Lagrangian describes interactions at a point or, because of the
derivative terms, neighboring points that are very close by. Therefore,
to obtain the propagator for a finite time interval it is necessary to
use the method of TIME SLICING. The idea is to break the time interval
up into N equal slices. In the limit N > ∞, the integral over positions
at each time slice can be said to be an integral over all possible paths.
The exponent becomes a timeintegral of the Lagrangian, namely the
action for each path.
Using the property that <ba> ≡ <an><nb> and <ba> = ∫b^{*}a dx we
can write:
<x_{f},t_{f}x_{i},t_{i}> = ∫ ... ∫<x_{i},t_{i}x_{1},t_{1}>dx_{1}<x_{1},t_{1}x_{2},t_{2}>dx_{2}<x_{2},t_{2}x_{3},t_{3}>
... dx_{N}<x_{N},t_{N}x_{f},t_{f}>
= ∫ ... ∫exp(iS(x_{i},t_{i};x_{1},t_{1})/h)dx_{1}exp(iS(x_{1},t_{1};x_{2},t_{2})/h)dx_{2}
... dx_{N}exp(iS(x_{N},t_{N};x_{f},t_{f})/h)
= ∫ ... ∫exp(iS(x_{i},t_{i};x_{1},t_{1})/h)exp(iS(x_{1},t_{1};x_{2},t_{2})/h)
... exp(iS(x_{N},t_{N};x_{f},t_{f})/h)dx_{1}dx_{2} ... dx_{N}
Therefore, to create a more general process involving motion it is
necessary to compound (multiply) individual actions, S as follows:
t x_{f}, t_{f}
 /
 / S
 o
 / S
 o
 / S
 o
 / S
 x_{i}, t_{i}

 x
Non Relativistic Free Particle

The Lagrangian for a free particle is:
.
L = mx^{2}/2
_{y} _{t} .
K(x,y) = ∫exp(i∫(x^{2}/2)dt)Dx with h = m = 1
_{x} _{0}
Splitting this into time slices yields:
K(x,y) = ∫Π_{t}exp{(i/2)([x(t + ε)  x(t))/ε]^{2}ε}Dx
If we replace the i with 1 the problem becomes easier to solve.
We will reinsert the i at the end. Doing this is analagous to a
Wick rotation (see below). Therefore,
K(x,y) = ∫Π_{t}exp{(1/2)([x(t + ε)  x(t))/ε]^{2}ε}Dx
Each factor in the product is a Gaussian integral centered at x(t)
with variance ε. The expression is a convolution of
Gaussians. K(x,y) can be represented by the
K(x,y) = G_{ε} * G_{ε} ... G_{ε}
The number of G's, n, is given by t/ε
If we take the Fourier transform the convolutions become
multiplications. Thus, of this we can write:
K(p) = G_{ε}(p)^{n}
The FT of G_{ε}(p) is another Gaussian:
G_{ε} = exp(εp^{2}/2)
Therefore,
K(p) = exp(tp^{2}/2)
The FT of this gives:
K(x,y) ∝ exp((x  y)^{2}/2t)
K(x,y) is a solution to the equation:
∂K/∂t = (1/2)∇^{2}K
Proof:
∂K/∂t = (x  y)^{2}exp((x  y)^{2}/2t)/2t^{2}
∂K/∂x = (x  y)exp((x  y)^{2}/2t)/t
∂^{2}K/∂x^{2} = (x  y)^{2}exp((x  y)^{2}/2t)/t^{2}
If we now insert the i back into the equation we get:
K(x,y) ∝ exp(i(x  y)^{2}/2t)
and,
∂K/∂t = (i/2)∇^{2}K
This is the Schrodinger equation with ψ replaced with K.
Extension to Fields

The FPI approach can be extend to fields by using the Lagrangian
density:
S[φ] = ∫d^{4}xL(φ,∂_{μ}φ)
Where the the 'paths' or histories being considered are not the motions
of a single particle, but the possible time evolutions of a field over
all space. The [φ] implies that the probability amplitude is computed
over all possible combinations of values that the field could have
anywhere in space–time. Thus, the field representation is a sum over
all field configurations, whereas the particle representation is a sum
over different particle paths. S[φ] is a functional of φ meaning that
the domain is no longer a region of space, but a space of functions that
are in turn functions of space and time.
The Lagrangian for the KleinGordon equation (real scalar field) is:
L = ∂_{μ}φ∂_{μ}φ  m^{2}φ^{2}
Therefore,
S[φ] = ∫d^{4}x (∂_{μ}φ∂_{μ}φ  m^{2}φ^{2})
and,
K(x,y) = ∫Dφ exp(iS[φ])
Where Dφ represents all possible field configurations in spacetime.
Therefore,
K(x,y) = ∫Dφ exp(i∫d^{4}x (∂_{μ}φ∂_{μ}φ  m^{2}φ^{2}))
This is a functional integral. Functional integration is a special
branch of mathematics that we will not concern ourselves with.
Instead, we will just state the result:
K(x,y) = ∫d^{4}p/(2π)^{4} exp(ip(xy))/(p^{2}  m^{2} + iε)
Which is the same result that we get from the canonical approach.
For a Fermionic field the situation is more complex and requires
the use of GRASSMANN ALGEBRA to derive the propagator. Grassmann
algebra is discussed in the section on supersymmetry.
Much of the current study surrounding QFT is devoted to the
properties of functional integrals, and how such integrals can be
made to be mathematically precise.
Lattice Gauge Theory

Lattice Gauge Theory is an alternate technique that overcomes the
challenges associated with functional integration and lends itself to
modern day computational methods.
In lattice gauge theory, the spacetime is Wick rotated into Euclidean
space and discretized into a lattice. A Wick rotation involves replacing
t by (it) in the Minskowki metric. When this is done the result is the
Euclidean metric.
Minkowski metric:
(ds^{2}) = (dt)^{2} + (dx)^{2} + (dy)^{2} + (dz)^{2}
Euclidean metric:
(ds^{2}) = (dτ)^{2} + (dx)^{2} + (dy)^{2} + (dz)^{2}
Problems are easier are generally easier to solve in Euclidean space
than they are in Minkowski space. The resulting solution may then,
under reverse substitution, yield a solution to the original problem.
Based on the concept of waveparticle duality, we can represent the
field in terms of the number of quanta at each value of (x,t) and the
number of quanta at each (x',t'). This is referred to as the FOCK/
CANONICAL representation of a quantum field. Therefore, φ now
consists of creation and annihilation operators.
We can the calculate the probability amplitude that the particle
content went from (x,t) to (x',t'). To do this, we construct a
lattice where each cell is associated with an action, S_{i}.
t
++++++++++ φ_{f}
         
++++++++++
          ^
++++++++++ 
          
++++++++++ history (analagous to trajectory)
         
++++++++++
         
++++++++++
         
++++++++++ φ_{i}
<final state S_{1}S_{2}S_{3} ... initial state>
To get the probability amplitude, P_{Γ} for the path replace
Π_{i}S_{i} with Π_{i}(1 + iS_{i}).
P_{Γ} can now be written as:
P_{Γ} = (1 + iS_{1})(1 + iS_{2})(1 + iS_{3}) ...
Now 1 + iS_{i} can be expanded as exp(iS_{i})
Thus, we can write after including h to match Feynman:
P_{Γ} = exp(i/h)S_{1}) exp(i/h)S_{2}) exp(i/h)S_{3})
= exp((i/h)Σ_{i}S_{i})
Now, S = ∫Ld^{4}x. For a very small volume, ε, we can approximate S as:
S = εL
Therefore, we can write:
P_{Γ} = exp((iε/h)Σ_{i}L_{i})
What kinds of things in the Lagrangian correspond to a particle
moving from one cell to another? The Lagrangian contains the
following derivatives:
(1/2)(∂φ/∂t)^{2} = (1/2){φ(x,t)  φ(x,t')/Δt}^{2}
(1/2)(∂φ/∂x)^{2} = (1/2){φ(x,t)  φ(x',t)/Δx}^{2}
As a result, the action (the ∫ of the Lagrangian over each cell)
will contain terms in φ(x,t) and φ(x',t'). As we stated before,
the ones of interest for motion are the crossterms φ(x,t)φ(x',t')
since they annihilated and create particles at different points.
If we just focus on these terms and ignore the rest for now we can
conceptually write:
P_{Γ} = exp(iΣφ(x,t)φ(x',t'))
Note that this is a greatly simplified treatment designed to
illustrate the basic concepts and is not rigorous. Now, the
exponential can be expanded as:
1 + iφ(x,t)φ(x',t') + (1/2!)Σφ(x,t)φ(x',t')Σφ(x',t')φ(x'',t'') ...
Consider:
++++
2   f  
++++
1    
++++
0  i   
++++
0 1 2
Such a process could be represented by:
1 + iφ(0,0)φ(1,0) + (1/2!)φ(1,0)φ(1,1)φ(1,1)φ(1,2)
or
1 + iφ(0,0)φ(0,1) + (1/2!)φ(0,1)φ(0,2)φ(0,2)φ(1,2)
The Σ's in the original expansion would act to sum all of the
possible paths to get from (0,0) to (1,2)
Of course, there are other terms in the Lagrangian that we haven't
included. For example there is the term m^{2}φ^{2}/2 that creates and
annihilates a particle at the same point and adds mass to the
otherwise massless processes describe above. For particles with
mass, the mass is included for weighting the probability amplitude
for the path.
We could also have a term like gφ^{3} which represents a 3 particle
interaction at a vertex. Similar to the above g, the coupling
constant, is included for weighting the probabilty amplitude for
the path.
In general, the Lagrangians can get extremely complicated and
difficult to solve.