Canonical quantization of scalar fieldsbased on S-3

Hamiltonian for free nonrelativistic particles:

Furier transform:

a(x) =!

d3p

(2!)3/2eip·x a(p)

we get:

can go back to x using:

!d3x

(2!)3eip·x = "3(p)

!d3p

(2!)3eip·x = "3(x)

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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 , eigenstate of with

is eigenstate of 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

Thus:

where is the lagrangian density.

is Lorentz invariant:

For the action to be invariant we need:

the lagrangian density must be a Lorentz scalar!

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Let’s consider:

Any polynomial of a scalar field is a Lorentz scalar and so are products of derivatives with all indices contracted.

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)

arbitrary constant

integration by parts, and at infinity in any direction (including time)

!"(x) = 0

is arbitrary function of x and so the equation of motion is Klein-Gordon equation

! = 1, c = 1

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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

where and are arbitrary functions of , and

is a function of (introduced for later convenience) |k|

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 is said to be on the mass shell)kµ

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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 k0 = !

for any the differential is Lorentz invariant

it is convenient to take for which the Lorentz invariant differential is:

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Finally we have a real classical solution of the K.-G. equation:

where again: , ,

For later use we can express in terms of :

where 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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!d3x

(2!)3eip·x = "3(p)

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