Speculations on Physics VI
This is the sixth installment of my speculations about physics, which occur sporadically in Noesis. For the record, the previous installments have been:
Noesis 43 (Nov 89): Schrödinger’s Cat and
Noesis 75 (Nov 92): Causality (with
Noesis 137 (Mar 98): Duality between Space, Objects, and Interactions
Noesis 138 (Sep 98): Octonions (with Robert Low)
Noesis 150 (Oct 00): Approximate Continuity
It is not well known outside of certain specialties in physics, but the major conservation laws are consequences of the major invariances. For example, conservation of energy is a consequence of the invariance of nature under time translation. Momentum conservation is a consequence of invariance under space translation. Angular momentum conservation is a consequence of invariance under rotation. In fact, it is possible to show that a continuous invariance leads to a conservation law. This is called Noether's Theorem.
To see how this works, let's consider time translation invariance. Time translation means moving the origin of time. Let's say we change coordinate systems from the old system with time t to the new system with time t'. The energy E of the system cannot change just because we changed coordinate systems, so we have
E(t') = E(t)
Now let's look at an translation d(t) for each moment t in the old coordinate system, such that
t' = t + d(t)
Then we have
E(t') = E(t + d(t))
If we take the limit of d(t) very small at all times t (a continuous invariance), then we have
E(t') = E(t + d(t)) = E(t) + (dE/dt) * d(t) = E(t)
(dE/dt) * d(t) = 0
for all possible forms of the infinitesimal d(t). From the calculus of variations we know that this is sufficient to conclude that
dE/dt = 0
for all values of t, or equivalently, that energy is conserved. QED.
If the universe is very large, then all points in it are nearly indistinguishable. Similarly, if the universe is very old, then all times are nearly indistinguishable. If the universe is nearly homogeneous, then all directions are nearly indistinguishable. This would seem to imply logically that if the universe is very large, then the laws of physics would be invariant under the time and space translation and rotation.
Consider the alternative. Suppose that the laws of nature were such that it is possible to detect a space transformation. This would seem to imply that there is some preferred location in space. Presumably this preferred location would be special in some way; perhaps it would be the "center" of the universe. But since the universe is very large, it should be very hard to detect this "center." And indeed the laws of nature as we know them now do not allow us to find any evidence of such a special place.
Thus it is possible that the invariance under time and space translation and rotation is a consequence of the universe being very large. Then by the theorems mentioned above, this implies that the various conservation laws will be observed in any universe that is sufficiently large.
This then suggests a line of inquiry. What would the universe be like if it were smaller? How would the departure from invariance show up in the laws of physics?
One way the universe could be smaller was discussed in the last installment. It may be that the universe is not composed of infinitely many points. If this were true, then perhaps this lack of scale invariance shows up as a needed change to the laws of physics. One would expect the new terms in the laws to be proportional to the fixed scale. Could this be the origin of the laws of quantum mechanics?