Transformations are fundamental

[rmirman]

This brings in the second condition. These transformations, like rotations, translations, Lorentz and Poincar\'{e} transformations and so on, are properties of geometry. As we will see requiring (as we must) that these are group basis vectors is often (always?) enough to determine the theory.

It is important to understand that these are transformation groups, not symmetry groups (although it is interesting that they are symmetry groups also). They are properties of geometry. These transformations relate observers (coordinates systems, objects). It is irrelevant whether geometry is invariant under them. It does not matter whether the states are stable, nor whether the Hamiltonian is invariant (there is no Hamiltonian). We consider observers (at a fixed time if there is some time dependence). Their observations are related by the transformations.

Thus (considering three dimensions) statefunctions are sums of the rotation group basis states, the spherical harmonics. If we rotate the rotated function is also such a sum. The terms in this sum are related to those of the first, with the coefficients depending on the angle. However each term is a sum of terms from the same rotation-group representation. A rotation changes the state but not the representation (total angular momentum). It is irrelevant whether geometry is invariant under rotations. Thus if we hide the symmetry by introducing a field (like gravity) it would seem that rotational invariance no longer holds. Nevertheless states of a representation go into states of the same representation (the projection of angular momentum is changed but not the total angular momentum). The basis states (spherical harmonics) are properties of the real three-dimensional geometry of our space. This would not be true if we used basis states of unitary groups.

There is another consequence of this. Geometry (the rotation group) only allows half-integral values of the angular momentum. No matter how badly the symmetry is broken there cannot be angular momentum values of 1/3 or ã.

Here we see that the functions (in quantum mechanics, statefunctions) are basis functions of the rotation-group representations and these are greatly restricted. Hence physical objects are. We shall find further, although less familiar, examples (OAIU, sec.~\ref{L}, p.~\pageref{L}) .

These results are not arbitrary. They are required. Neither God nor nature can challenge them. Geometry is fundamental. Physics takes place in geometry so geometry rules. This might be true of all of physics, although we do not know that yet.

Dirac's equation

Why does Dirac's equation hold? Despite an all too prevalent belief it is not some strange property of nature. It is not because God likes it. It is a trivial property of geometry. Considering only space transformations, ignoring interactions and internal symmetry, objects (thus free) belong to states of the Poincar\'{e} group. This has two invariants (like the rotation group has one, the total angular momentum). For a massive object these are the mass and spin in the rest frame. Knowing these the object is completely determined (since interactions and all other parameters are ignored). For massless objects, thus with one invariant determined, we have another, the helicity. Thus two equations, not one, are needed to determine an object. For spin-${1 \over 2}$, only, these two can be replaced by one, Dirac's equation. Why is this? The momentum, $p_{\mu}$ is a four-vector. There is another four-vector, $\gamma_{\mu}$. Thus $\gamma_{\mu}p_{\mu}$ is an invariant. It is a property of the object, and we give that property the name mass (which is the real meaning of mass). Thus

\begin{equation}\gamma_{\mu}p_{\mu} = m, \end{equation}

which is Dirac's equation. It gives the mass of the object, and the spin, ${1 \over 2}$. This is only possible because of the $\gamma_{\mu}$'s. These form a Clifford algebra and there is (up to inversions) only one for each dimension. This is then the reason for Dirac's equation, and only for a single spin.

Schr”dinger's equation is its nonrelativistic limit, and Newton's second law the classical version of that. That is why these hold. Hamiltonian's equation, for example, is just another version of Newton's second law.

Although two equations are needed, except in the special case of spin-${1 \over 2}$, classical objects consist of collections of objects with this spin, ${1 \over 2}$, protons, neutrons and electrons. That is why Newton's second law holds classically.

Notice that these are a property of geometry. The Poincar\'{e} group is the transformation group of our real, 3-dimensional geometry, the ˜'s a property of geometry, and the dot (scalar) product is determined by geometry. Thus basic equations of our world are purely geometrical. We shall see that electromagnetism and gravitation are also purely geometrical (MRPG).

What about interactions? We know how to put them in. We start with the Hamiltonian for which interactions are nonlinear terms. We must pick the right ones. There are restrictions but still much freedom; much remains unknown. The Hamiltonian is one of the four components of the momentum, and these are transformed into each other (by geometrical) 4-rotations (Lorentz transformations). This gives the momentum operators with interactions. With these we can get Dirac's equation with interactions. (It is probably true, but not completely certain that we can put interactions into Dirac's equation, which we can do with the electromagnetic interaction, but which has not been proven to be true in general.)

Of course this leaves out internal symmetry, which remains a fundamental mystery (and which is not completely symmetric).

Quantum field theory

What is quantum field theory and how does it differ from quantum mechanics? Physical objects (statefunctions) must be basis functions of the transformation group of space. As we know from the rotation group these can be products of basis functions. (However for the rotation group such products can be reduced into a sum of basis functions, which may not be true for more complicated groups like the Poincar\'{e} group.) There is fundamentally no difference between such basis states that are products of other basis states and single basis states. For products we have equations for each basis state in the product, but can often write just a single equation for the product. In general we can write an equation which is a sum of (an infinite number of) terms for each object in the product, with interaction terms transforming terms (many appearing 0 times) into others.

This can also be written as a set of equations, each for a single term, with interactions related them, a set of coupled equations.



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

[mkirkpatrick]

Thanks rmirman,There is indeed much tranforming taking place,I have yet to find anything within this physical universe the is more economical in the energy tranference
business as that of the vortex.

warm regards michael.