Wheeler, Mechanics

2.4 Functional integration
It is also possible to integrate functionals. Since a functional has an entire function as its argument, the
functional integral is a sum over all functions in some well-defined function space. Such a sum is horribly
uncountable, but it is still possible to perform some functional integrals exactly by taking the infinite limit
of the product of finitely many normal integrals. For more difficult functional integrals, there are many
approximation methods. Since functional integrals have not apppeared widely in classical mechanics, we do
not treat them further here. However, they do provide one approach to quantum mechanics, and play an
important role in quantum field theory.
3 Physical theories
Within any theory of matter and motion we may distinguish two conceptually different features: dynamical
laws and measurement theory. We discuss each in turn.
By dynamical laws, we mean the description of various motions of objects, both singly and in combination.
The central feature of our description is generally some set of dynamical equations. In classical mechanics,
the dynamical equation is Newton’s second law,
F = m dv
dt
or its relativistic generalization, while in classical electrodynamics two of the Maxwell equations serve the
same function:
1 dE
c dt − ∇ × B = 0
1 dB
c dt + ∇ × E = 4π c J
The remaining two Maxwell equations may be regarded as constraints on the initial field configuration. In
general relativity the Einstein equation gives the time evolution of the metric. Finally, in quantum mechanics
the dynamical law is the Schrödinger equation
Ĥψ = i ∂ψ
∂t
which governs the time evolution of the wave function, ψ.
Several important features are implicit in these descriptions. Of course there are different objects –
particles, fields or probability amplitudes – that must be specified. But perhaps the most important feature
is the existence of some arena within which the motion occurs. In Newtonian mechanics the arena is Euclidean
3-space, and the motion is assumed to be parameterized by universal time. Relativity modified this to a
4 -dimensional spacetime, which in general relativity becomes a curved Riemannian manifold. In quantum
mechanics the arena is phase space, comprised of both position and momentum variables, and again having
a universal time. Given this diverse collection of spaces for dynamical laws, you may well ask if there is any
principle that determines a preferred space. As we shall see, the answer is a qualified yes. It turns out that
symmetry gives us an important guide to choosing the dynamical arena.
A measurement theory is what establishes the correspondence between calculations and measurable numbers.
For example, in Newtonian mechanics the primary dynamical variable for a particle is the position
vector, x. While the dynamical law predicts this vector as a function of time, we never measure a vector
directly. In order to extract measurable magnitudes we use the Euclidean inner product,
〈u, v〉 = u · v
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