A field configuration (or field equations) are "symmetric" under a "transformation" when the transformation leaves the configuration (or form of the equations) unchanged. For example, a field is "symmetric with respect to rotations in three dimensions" or "spherically symmetric" if it is unchanged when the spatial coordinates are rotated in any direction. (Note that this implies that absolute angle is not a measurable quantity.) Numerical quantities which remain unchanged under such transformations are "conserved". When a set of transformations is closed (any transformation can be expressed as the product of other transformations in the set), the set is called a "symmetry group". Groups of interest in Quantum Gravity are:
| Group | Transformation | Unmeasurable Quantities | Conserved Quantities |
|---|---|---|---|
| Rotation (SO(3)) | Spatial Rotations | Absolute Angle | Angular Momentum (L) |
| Translation | Spacetime Translations | Absolute Position | Energy (E) or Mass (M) and Momentum (P) |
| Lorentz | Spacetime Rotations and Reflections | Absolute Uniform Velocity, Orientation | Spacetime Interval (S), Parity (P) and Time Reflection (T) |
| SL(2,C) ("Homogeneous Lorentz") | Spacetime Rotations | Absolute Uniform Velocity | S (not P or T) |
| Diffeomorphism ("General Coordinate") | Smooth curving of spacetime (Accelerations) | Absolute Acceleration | Topological Invariants* |
| Poincare | Lorentz + Translations | (see above) | L, E (or M) and P |
| U(1) | Scalar Phase Shift | Absolute Phase | Electric Charge |
| SU(2) | 2D Phase Shift | Absolute 2D Phase | Isospin |
| SU(3) | 3D Phase Shift | Absolute 3D Phase | Color |
* Note that the requirement that diffeomorphisms be invertible induces a differential structure on the manifold in question. Manifolds with multiple differential structures (like the 7-dimensional sphere) have multiple, mutually-exclusive equivalence classes of metrics, which are characterized by different, independent definitions of volume (since the volume form changes by a factor of the Jacobian under coordinate transformations).
Quantum Field Theory as implemented in the Standard Model of Particle Physics makes extensive use of the last four of the above groups. All fermionic field quanta "carry representations" of those groups: the quanta have definite properties under transformations in those groups, identified by numerical charges. Hence an "up quark" quantum has a definite mass (400 MeV) and "spin" or intrinsic angular momentum (one half) associated with the Poicare Group; also a definite electric charge (2/3 e), isospin ("up") and color (red, green or blue, which are designations for the three degrees of freedom and have nothing to do with visible light), associated with the "unitary" groups U n. In comparison, a "down quark" has the quantum numbers 700 MeV, one half, -1/3 e, "down" and R, G or B. The other elementary fermionic field quanta are the electron (.5 MeV, one half, -1, "up") and the neutrino (0 MeV (?), one half, 0, "down") (both called "leptons"), which are colorless and hence do not transform under SU(3) color (alternatively, they are called color "singlets"). When the angular momentum of any of these femionic quanta is parallel to their momentum they are "left-handed" and transform as an "isospin doublet" (the up and down quarks form a doublet, as do the electron and neutrino). When they are right-handed, they are isospin singlets. Note that the handedness or "helicity" can be changed by a Lorentz transformation, but only for massive quanta. Quanta of positive energy modes are associated with particles and those of negative energy modes with "antiparticles". Antiparticles have the same mass but opposite charge(s) and helicity of their particle counterparts. Only left-handed neutrinos and right-handed antineutrinos have been observed.
The elementary interactions between fermionic field quanta are modelled as mediated by "gauge bosons": bosonic field quanta which carry spin one representations of their respective groups. Hence the electromagnetic force is the result of the exchange of a photon, the U(1) gauge boson; the "weak" force is the result of the exchange of SU(2) isospin bosons (which change, for example, an up quark into a down quark) and the "strong" force is the result of the exchange of SU(3) color "gluons" (which change quark colors and in doing so bind quarks together). The weak bosons carry electric charge and are massive (in constrast to the photon and gluons). The weak gauge bosons may interact among themselves (as may the strong), since they carry representations of the group they mediate (weak bosons have isospin and gluons have color). Their respective groups are called "non-Abelian" in contrast to U(1), which is "Abelian" (photons cannot interact directly). The use of gauge bosons as intermediaries in nonlocal interactions serves to restore the local nature of the theory.
The parameter space of the "unitary" groups (U(1), SU(2) and SU(3)) are "isomorphic to" (have a one-to-one correspondence with) the circle, the sphere (a surface!) and CP2 (complex projective space in 2 dimensions). Note that the fields associated with the nonabelian symmetries can be divided into "electric" (curl-free) and "magnetic" (divergence-free) fields just as in the the abelian case (electromagnetism).
Quantum Gravity Concept Map Index:
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©1997, Kenneth R. Koehler. All Rights Reserved.