Re: definition of field

[John Denker <jsd@MONMOUTH.COM>]

At 03:29 PM 5/22/99 -0700, Peter Vajk wrote:
> As I recall from my undergraduate physics a long time ago, "field" was
> defined as "force per UNIT (mass/charge/hadron number, etc.

Hmmm, that is not the *defining* property of a field.  (It does perhaps
call attention to an interesting *secondary* property of the fields, as
discussed below.)

I tend to prefer statements such as this one from the Encycopedia Britannica:
        A field in physics may be defined as a continuous
        distribution of some observable quantity in space and time.

Examples include:
a) A fluid has a temperature field.  That is, the fluid has a temperature
at each point in space.
b) A fluid has a velocity field.  That is, the fluid has a velocity at each
point in space.

The thing we call the electric field is a field -- not because it is "per
unit something", but because it has a value at each point in space (and time).

The terminology is not 100% consistent, because of the following broken
pattern:
a) a temperature field, evaluated at a given point, is a temperature.
b) a velocity field, evaluated at a given point, is a velocity.
c) an electric field, evaluated at a given point, is an electric?  Oops.

We observe that in example (c), the word electric is an adjective
describing the origin of the field, not directly describing its field
properties.

There are various non-broken things we can say, including:
a) a temperature field, evaluated at a given point, has dimensions of
temperature.
b) a velocity field, evaluated at a given point, has dimensions of velocity
c) an electric field, evaluated at a given point, has dimensions of force
per unit charge.
d) a gravitational field, evaluated at a given point, has dimensions of
acceleration.

=========

We now turn to the secondary point:  Whereas science-fiction movies speak
of force fields, it would be quite problematic to have such a thing,
strictly speaking.
  field of (temperature), OK
  field of (velocity), OK
  field of (force per unit mass), OK
  field of (force per unit charge), OK
  field of (force per unit volume) (i.e. buoyancy), OK
but
  field of force per se, not OK

The basic reason is that force is an extensive quantity.  You can't have an
extensive quantity at each infinitesimal point in space.  So it is not an
accident that force-related fields are always force per unit something.
But the other way around it doesn't work:  force per unit something does
not imply (let alone define) the existence of a field.

==========

As if there weren't enough confusion already, let me call attention to one
more source of confusion:

The term "field" has a well-established definition in mathematics (as in
groups, rings, fields, et cetera).  The rational numbers (R) are a field.
The real numbers are a field.  The complex numbers are a field, which can
be considered a vector field since the complex numbers are isomorphic to R^2.

Ordinary 3-dimensional location-space is a vector space.  It is not a field
in the mathematical sense.  There's no suitable multiplication operator.
The cross product won't do, since there's no multiplicative identity.

None of the aforementioned physical fields is a field in the mathematical
sense.  Sorry.

Mathematicians might say that the electrical interaction can be described
by a vector-valued function on a vector space, that is, at each point in
location-space there is a vector space (described by the components of E at
that point).

We probably don't need to go into the details on this;  the take-home
message is that you should be careful if somebody comes to you with an
axiomatic definition of "field".

Cheers --- jsd