[rmirman]

What determines the laws of nature? Perhaps God? Physicists seem to accept them as given, with no rationale, purely arbitrary and accidental (which is perhaps equivalent to God). Yet there are almost certainly reasons for them, and for many these we know. Whether God likes it or not these must be the laws governing the universe.

Here we consider two requirements that turn out to be extremely powerful, perhaps completely determinative. These are consistency and geometry. Physical laws must of course be consistent. This is so obvious that it is not generally realized how powerful and subtle it is, and how difficult to use. Laws may seem correct but only because not all conditions are included or understood.

The dimension of space must be 3+1 because physics would be inconsistent, impossible, in any other dimension. There could be no universe (OAIU, sec.~\ref{L}, p.~\pageref{L}). This emphasizes that geometry imposes (strong) conditions on physics, but also physics imposes strong conditions on geometry. Only in some (one?) geometries is physics possible.

Classical physics is inherently inconsistent. Quantum mechanics necessary. Neither nature or God can disagree, mathematics rules. What is wrong? Consider Newton's second law, properly written (OAIU, sec.~\ref{L}, p.~\pageref{L}). How could it be impossible? It illustrates the importance of language and how it so confuses and misleads. The law contains a dangerous word: force. What is a force? Friction, normal force? But there are no such things. They are purely phenomenological representations, introduced for tractability, for the actual physical object: electromagnetism (and similarity for gravitation). But an electromagnetic field is a sum of terms of the form (all equations are phenomenological), exp(ikx + ift). Thus on one side of the equation we have a real function of time, on the other a complex function of both space and time. Newton's second law is nonsense. Actually so is the concept of an electromagnetic field (MRPG).

The problem is the use of quantities like position and momentum. These (which correctly are quantum mechanical expectation values) are the wrong quantities. Physics cannot be built on them.

We must use functions, ones of both space and time, complex functions. And this is quantum mechanics (there is really no difference between quantum mechanics and quantum field theory). These are statefunctions (a better term than wavefunctions since nothing waves).

(We see again that language confuses. A worse example is quantum mechanics, implying discreteness. That is neither fundamental nor universal. Free particles and those going through a barrier have continuous energies; classical waves in a waveguide are discrete as are violin strings. Discreteness comes from boundary or regularity conditions. Yet because a bad name is used people --- physicists! --- are completely confused. Language is dangerous. Something like functional mechanics would be better.)

There is a fundamental requirement on these functions: they must be basis vectors of the transformation groups of space: the rotation group which is a subgroup of the Lorentz group, itself a subgroup of the Poincar\'{e} group (a subgroup of the conformal group --- QFT,CGT,CFT).

Why, what does this mean? There are different observers, relatively rotated, translated, with different velocities. These are not people, humans are not nearly as important as we like to believe, but objects, all objects (GTFQM, particularly the cover). Their observations, statefunctions, must be related, and they are related by the transformations forming groups. Different coordinate systems --- observers --- are related by geometrical transformations, rotations or translations say. Hence it must be possible for the groups to transform them. Otherwise there would be not relationship between different observers, even infinitesimally different (OAIU, sec.~\ref{L}, p.~\pageref{L}). Physics would be inconsistent, not possible. This means (by definition) that they are representation basis functions of the group. These are (sums of) plane waves or spherical harmonics, for example.



OAIU;
Our Almost Impossible Universe:
Why the laws of nature make the existence of humans extraordinarily unlikely

GTFQM;
Group Theoretical Foundations of Quantum Mechanics

MRPG;
Massless Representations of the Poincaré Group

QM,QFT;
Quantum Mechanics, Quantum Field Theory
geometry, language, logic

QFT,CGT,CFT;
Quantum Field Theory, Conformal Group Theory, Conformal Field Theory:

GT:IA:
Group Theory: An Intuitive Approach

PG,SG;
Point Groups, Space Groups, Crystals, Molecules