Download Canonical quantization of scalar fields

Canonical quantization of scalar fields
based on S-3
Hamiltonian for free nonrelativistic particles:
Furier transform:
a(x) =
we get:
!
d3 p
ip·x
e
ã(p) !
3/2
(2π)
d3 x
(2π)3
eip·x = δ 3 (p)
can go back to x using:
!
d3 p ip·x
3
e
=
δ
(x)
3
(2π)
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Canonical quantization of scalar fields
(Anti)commutation relations:
[A, B]∓ = AB ∓ BA
Vacuum is annihilated by
:
is a state of momentum
is eigenstate of
, eigenstate of
with
with energy eigenvalue:
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Relativistic generalization
Hamiltonian for free relativistic particles:
spin zero, but can be either bosons or fermions
Is this theory Lorentz invariant?
Let’s prove it from a different direction, direction that we will use for
any quantum field theory from now:
start from a Lorentz invariant lagrangian or action
derive equation of motion (for scalar fields it is K.-G. equation)
find solutions of equation of motion
show the Hamiltonian is the same as the one above
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A theory is described by an action:
where
is the lagrangian.
Equations of motion should be local, and so
where
is the lagrangian density.
Thus:
is Lorentz invariant:
For the action to be invariant we need:
the lagrangian density must be a Lorentz scalar!
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Any polynomial of a scalar field is a Lorentz scalar and so are products of
derivatives with all indices contracted.
Let’s consider:
arbitrary constant
! = 1, c = 1
and let’s find the equation of motion, Euler-Lagrange equation:
(we find eq. of motion from variation of an action: making an infinitesimal
variation
in
and requiring the variation of the action to vanish)
integration by parts,
and δφ(x) = 0 at
infinity in any direction
(including time)
is arbitrary function of x and so the equation of motion is
Klein-Gordon equation
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Solutions of the Klein-Gordon equation:
one classical solution is a plane wave:
is arbitrary real wave vector and
The general classical solution of K-G equation:
where
and
are arbitrary functions of , and
is a function of |k| (introduced for later convenience)
if we tried to interpret
as a quantum wave function, the second term
would represent contributions with negative energy to the wave function!
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real solutions:
k −→ −k
thus we get:
(such a
k µ is said to be on the mass shell)
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Finally let’s choose
so that
is Lorentz invariant:
manifestly invariant under orthochronous Lorentz transformations
on the other hand
sum over zeros of g, in our case the only zero is k 0 = ω
for any
the differential
it is convenient to take
the Lorentz invariant differential is:
is Lorentz invariant
for which
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Finally we have a real classical solution of the K.-G. equation:
where again:
For later use we can express
where
,
,
in terms of
:
and we will call
Note,
.
is time independent.
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Constructing the hamiltonian:
Recall, in classical mechanics, starting with lagrangian
as a function
of coordinates
and their time derivatives
we define conjugate
momenta
and the hamiltonian is then given as:
In field theory:
hamiltonian density
and the hamiltonian is given as:
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In our case:
Inserting
we get:
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!
d3 x ip·x
e
= δ 3 (p)
3
(2π)
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