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From: brooks@lll-crg.ARPA (Eugene D. Brooks III)
Newsgroups: net.physics
Subject: Re: [brooks: Quantum Field Theory]
Message-ID: <738@lll-crg.ARPA>
Date: Sat, 27-Jul-85 22:17:18 EDT
Article-I.D.: lll-crg.738
Posted: Sat Jul 27 22:17:18 1985
Date-Received: Wed, 31-Jul-85 00:58:45 EDT
References: <439@sri-arpa.ARPA>
Organization: Lawrence Livermore Labs, CRG group
Lines: 27

> I would be very interested to hear an explanation of how the
> four dimensional space we live in is defined in terms of quantum field
> theory.  Simple quantum mechanics would do since I am more familiar
> with that.  Furthermore we can ignore gravity;  I am willing to assume
> that space-time is flat.  Eventually I would like to understand the
> "last principle" derivations.

In quantum field theory, the four dimensional space we live in (x,y,z,t)
are scalar indeces of the field operators.  The position in space-time is
not an operator as it is in simple quantum mechanics.  One expresses a
Hamiltonian (energy) operator in terms of the field operators.  There
are also corresponding momentum operators Px, Py and Pz which are expressed
in terms of the field operators.
If one defines the initial conditions on the field operators at time t=0 then
the Hamiltonian is used to propagate the initial state of the operators
using the standard Heisenberg picture Schroodinger equation for the field
operators.

The specification of the field operators on the t=0 surface of 4D space time
is a non-Lorentz invariant element of the theory.  This can also be gotten
rid of by specifying the initial conditions on any space like surface.
If you do this the evolution of the system runs along a normal to this
surface.  The generalized Hamiltonian (Px, Py, Px, H) is used to do the
propagation of the initial specification of the field operators.

See: "Relativistic Quantum Fields", Bjorken and Drell
     "Quantum Field Theory", Itzykson and Zuber