Riff-in-Thyme said:
Can that function space be defined without reference to a particle?
Here’s the way I see it (correctly or incorrectly):
What you are asking about is a quantum field theory. Physics provides two distinct ways to describe reality: mechanics and field theory. Mechanics describes reality in terms of particles, describing their position in spacetime in terms of a single parametric variable:
x = x(s)
y = y(s)
z = z(s)
t = t(s)
where s is the parameter. Although one usually encounters three equations describing spatial position in terms of the time variable, the above places all spacetime variables on equal footing as required by relativity.
By contrast, field theory describes reality in terms of a field function over all spacetime variables:
F = F(x,y,z,t)
Note the difference: in mechanics, the spacetime variables are the dependent variables, whereas in field theory, they are the independent variables. Field theory is the more powerful description as it can readily describe particles, but mechanics can’t readily describe fields. Also, field theory automatically places all spacetime variables on equal footing without the need to invoke an extra parameter.
Physics also describes two distinct realms: classical and quantum. Combined we have the four combinations:
Classical Mechanics
Classical Field Theory
Quantum Mechanics
Quantum Field Theory
Here’s the difficulty: the nature of quantum physics demands that quantum mechanics be described as if it were a classical field theory. So how is quantum field theory to be described? My solution (not necessarily the standard solution) is to force quantum mechanics into a mechanical description. Then quantum field theory becomes the corresponding field theoretical description. However, the resulting quantum mechanics must still be equivalent to a classical field theory. This is done by extending the domain from spacetime to a function space. Then, classical field theory (quantum mechanics) becomes a point in this space, and quantum field theory becomes a field function over this space. In other words, reality is being described as a quantum superposition of entire spacetimes.