Foundations of Quantum Field Theory. Klaus D Rothe

Foundations of Quantum Field Theory

with n standing for a discrete or continuous label.

(4)Observables

To every observable corresponds a hermitian operator; however, not every hermitian operator corresponds to an observable.

(5)Symmetries

Symmetry transformations are represented in the Hilbert space

by unitary (or anti-unitary) operators.

(6)Vector space

The complete system of normalizable states |Ψ

defines a linear vector space.

(7)Covariance of equations of motion:

and

′ denote two inertial reference frames, then covariance means that the equation

implies

Furthermore, there exists a unitary operator U which realizes the transformation

→

′:

(8)Physical states

All physical states can be gauged to have positive energy¹

(9)Space and time translations

Space-time translations are realized on |Ψ

respectively by²

where

with

the generator of rotations

1.2Principles of NRQM not shared by QFT

The following principles of non-relativistic quantum mechanics must be abandoned in the case of QFT:

(1)Probability amplitude

In NRQM we associate with the state |Ψ(t)

a wave function

then represent the probability of finding a particle in the interval

at time t. (Notice the treatment of space and time on unequal footing.) In QFT we can have particle production, that is, we are dealing with “many-particle” physics. Hence notions linked to a one particle picture must be abandoned in the relativistic case.

(2)Galilei transformations

For a scalar function

and a Galilei transformation

, t′ = t we must have

Consider in particular a plane wave

in S as seen by an observer in S′ moving with a velocity

with respect to S:

where

We have

with

We seek an operator

with the property

From

and

we conclude, by comparing with (1.1),

where

is the position operator. Now

Hence we may write

in the form

where we have used

Denoting by

the eigenstates of the momentum operator

we have

For the solution

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