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      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

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

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