Foundations of Quantum Field Theory. Klaus D Rothe

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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

      (2)Time development

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

      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

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

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