[Phys-L] Re: What IS a field? (Was Re: Physics (chem.) Humor)

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding Bernard's question:

> Is space empty when it contains fields.  How is matter different
> from its fields?

The fields exist even when their values locally happen to be zero.
On a classical level the fields are number-valued (generalized)
functions of the events of space-time. For each  event in space-time
there is a numerical value for each component of each field present.

But on the quantum level these fields are promoted to being mappings
from the events of space-time to the space of linear operators in
Hilbert/Fock space.  IOW for each event in space time and for each
component for each field there is its own corresponding operator in
Fock space (rather than just its own value of a number) for each
field component present.

Space is as empty as it can be when the fields have no excitations
and only take on quantum zero point fluctuations.  That state is
said to be the vacuum.  The particles of matter and radiation exist
as quantum excitations of the corresponding fields.  These
excitations are quantum states that have been created from the
vacuum state (& sometimes partially destroyed from other more
complicated states) by the creation and destruction terms in the
corresponding field operators.

And Rodney's questions:

> The more I learn about fields the more confused I am about them.  Do
> the particles create the fields or do the fields create the
> particles?

The creation terms in the field operators create the states that are
interpreted as excitations, i.e. particles.

> Does that question even make sense?  What IS a field?

Yes, I think so.  A field is mathematically represented as a
linear operator-*valued* distribution (generalized functional
mapping in the Lebesgue/Stiltjes measurability sense) on the events
of the space-time manifold at the quantum level.  On the classical
level a field is a generalized function (w.r.t. the Lebesgue/Stiltjes
measure allowing for delta-function-type limits of function
sequences) on the events of the space-time manifold.

> I do tell my students that an electric field (for example) is a real
> thing, just as real as the charged particle that creates it.  But
> thankfully, no one has asked me to elaborate on "create."

That's fortunate.  Yeah, it's a pain to have to elaborate on a
mouthful of technical terminology. :-)

> --
> Rodney Dunning
> Assistant Professor of Physics
> Birmingham-Southern College

David Bowman
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