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Chapter 1

The Principles of Quantum Physics

Quantum Field Theory is a natural outgrowth of non-relativistic Quantum Mechanics, combining it with the Principles of Special Relativity and particle production at sufficiently high energies. We therefore devote this introductory chapter to recalling some of the basic principles of Quantum Mechanics which are either shared or not shared with Quantum Field Theory.

1.1Principles shared by QM and QFT

We briefly review first the principles which non-relativistic Quantum Mechanics (NRQM), relativistic Quantum Mechanics (RQM) and Quantum Field Theory (QFT) have in common.

(1)Physical states

Physical states live in a Hilbert space _phys and are denoted by |Ψ.

(2)Time development

In the Schrödinger picture, operators OS are independent of time and physical states |Ψ(t) obey the equation,

with H the Hamiltonian.

In the Heisenberg picture physical states |Ψ_H are independent of time and operators O(t)_H obey the Heisenberg equation

The states in the two pictures are related by the unitary transformation

(3)Completeness

Eigenstates |Ψ_n > of H,

are assumed to satisfy the completeness relation

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:

If 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 is the momentum operator, and rotations are realized on |Ψ by

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

of the “free” Schrödinger equation we obtain from (1.3),

or

Furthermore, making use of

we have

or

with

Correspondingly we have from (1.7)

in accordance with expectations.

(3)Covariance of equations of motion

From

follows

which we rewrite as

Noting from (1.3) and (1.2) that

we obtain, using (1.6),

or, recalling (1.7),

This equation expresses on operator level the covariance of the free-particle equation of motion: If is a solution of the equations of motion, then is also a solution.

Group-property

As Eq. (1.4) shows, a Galilei transformation is represented by the unitary operator

with

the generators of boosts. We have

so that different “boosts” commute with each other. The Galilei transformations thus correspond to an abelian Lie group. In particular

Boosts

Let S and S′ be two inertial frames whose clocks are synchronized in such a way, that their respective origins coincide at time t = 0. Then we have for an eigenstate of the momentum operator, as seen by observers O and O′ in S and S′

respectively. In particular, for a particle at rest in system S we obtain, from the point of view of O′,

Define

where stands for a Galilei transformation taking and . U[B()] is thus an operator which takes a particle at rest into a particle with momentum . One refers to this as a “boost” (active point of view).

We have the following property of Galilei transformations not shared by Lorentz transformations (compare with (2.23)): boosts and rotations separately form a group. Indeed one easily checks that

(4)Causality

We next want to show that NRQM violates the principle of causality. We have for any interacting theory,

Let |En be a complete set of eigenstates of H:³

Then

Define

as well as

The kernel satisfies a heat-like equation:

In terms of this kernel we have from above,

We now specialize to the case of a free point-like particle. In that case

and correspondingly we have with (1.8),

Notice that the kernel K₀ satisfies the desired initial condition (1.9).

From (1.10), for the initial condition , we get

or in particular

that is, for an infinitesimal time after t₀ one already finds the particle with equal probability anywhere in space; this violates obviously the principle of relativity, as well as causality.

________________________

¹In a relativistic theory, can also contain negative energy states, which then require a particular interpretation or must decouple altogether from the “physical sector” of the theory.

²In Quantum Mechanics

In this chapter we use everywhere lower indices, repeated indices being summed over.

³For notational simplicity we suppose the spectrum to be discrete.