Invariance mechanics
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Invariance mechanics, in its simplest form, is the rewriting of the laws of quantum field theory in terms of invariant quantities only. For example, the positions of a set of particles in a particular coordinate system is not invariant under translations of the system. However, the (4-dimensional) distances between the particles is invariant under translations, rotations and Lorenz transformations of the system.
The invariant quantities made from the input and output states of a system are the only quantities needed to give a probability amplitude to a given system. This is what is meant by the system obeying a symmetry. Since all the quantities involved are relative quantities, invariance mechanics, can be thought of as taking relativity theory to its natural limit.
Invariance mechanics has strong links with loop quantum gravity in which the invariant quantites are based on angular momentum.
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[edit] Feynman Rules
The Feynman rules of a quantum system can be rewritten in terms of invariant quantities (plus constants such as mass, charge, etc.) The invariant quantities depend on the type of particle, scalar, vector or spinor. The rules often involve geometric quantities such as the volumes of simplices formed from vertices of the Feynman graphs.
[edit] Scalar Particles
In a system of scalar particles, the only invariant quantities are the 4-dimensional distances (intervals) between the starting points (x) and ending points (x') of the particle paths. These points form a complete graph:
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The invariants are the numbers
- R = | x − x' |
[edit] Vector Particles
In a system of vector particles such as photons, the invariants are the 4-dimensional distances between the starting points and ending points of the particle paths, and the angles between the starting and ending polarisation vectors of the photons ( ρμ )
There are 4 invariants on each line:
- R = | x − x' |
- S = ρμρ'μ
- T = (x − x')μρ'μ
- U = ρμ(x − x')μ
[edit] Yang-Mills Vector Particles
Yang-Mills vector fields of a given gauge group also involve the angle representing a rotation of the gauge group ( 
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There are 3 invariants on each line:
- R = | x − x' |
[edit] Spinor Fields
These involve the angles between the spinor vectors. The invariants are:
- R = | x − x' |
[edit] Mixed systems
Systems usually consist of a mixture of scalar, spinor and vector fields and the invariants can depend on angles between spinors and vectors. To simplify this process ideas from Twistor theory are often used.
[edit] Constraints
A system represented by a complete graph contains many invariant quantities. For large graphs, however, not all these quantities are independent and we must specify dimensional and gauge constraints. Why the particular number of dimensions or particular gauge group is chosen is still not known. The constraints and whether they are satisified exactly or approximately is the key to invariance mechanics and the difference between it and conventional field theory. Work is being done to see whether the breaking of these constraints is a consequence of the gravitational field.
[edit] Dimensions
Since invariance mechanics does not explicity use coordinate systems, the definition of dimension is slightly different. The equivalent way of expressing the number of dimensions is given, as in distance geometry, as specifying that the volume of any (D+2)-simplex made from the points in the system is zero. The volume of a simplex is given by a formula involving the invariant distances (the R's) between the points which is given by the Cayley-Menger determinants. If this determinant is exactly 0 for all simplices then the geometry is Euclidean. If the determinant is only approximately 0 then at small distances space-time is non-Euclidean. This has deep connections with quantum foam and loop quantum gravity.
[edit] Gauge Group
In a similar way to expressing the number of dimensions, the dimension and type of the gauge group is given by an identity involving the polarisation (or spin) invariants (the S, T and U's). The gauge group is an internal symmetry because the gauge identity involves far more quantities than the dimensional identity. There has been recent work on combining the dimensional and gauge constraints into a single equation to produce a unified theory.
