Saturday, February 05, 2005

Why spontaneous symmetry breakdown in the vacuum creates a preferred Lorentz frame

Paul, first think of the ferromagnet in ordinary 3D Galilean relativity Euclidean space. The physical dynamics is rotationally invariant under the 3D orthogonal rotation group O(3) that leaves the quadratic form invariant, i.e.

x^2 + y^2 + z^2 = x'^2 + y'^2 + z'^2

Where there is a matrix representation M of the elements of the group that perform a linear transformation

(x',y'z') = M(x,y,z)

Alternatively, think of two orthonormal frames of reference (ex,ey,ez) (ex',ey',ez')

The fixed 3-vector r is then, in the two base state (frames of reference)

r = xex + yey + zez = x'ex'+ y'ey' + z'ez'


(ex',ey',ez') = M^-1(ex,ey,ez)

The adjoint M* = M^-1 over the field of real numbers is the definition of the orthogonal groups.

Therefore, in this linear world of 3D space flat Euclidean geometry, we think of O(3) as transformation connecting different triads (or more generally frames of reference relative to the group) oriented differently to each other. The dynamical laws of nature must be "tensors" relative to the group O(3). That means, that the laws of nature must be O(3) covariant if we apriori assume that physical space is isotropic. Right now we think of an affine space without any physical origin and the triads (base sets) are global reference frames, i.e. the unit vectors point in definite directions and trivially parallel sets of triads are in an equivalence relation that we factor out for now. An O(3) tensor transforms multi-linearly.

So what happens in a ferromagnet? A particular class of triad frames (ex,ey,ez) is physically selected or preferred in a finite region "domain" of space. This means that there is a local order parameter, here the magnetization in the ground state of the material that points in a definite direction at every location in the domain. Let ez be the direction of this actual magnetization. Of course ex & ey can rotate around ez described by O(2) rotations forming an equivalence relation in the plane perpendicular to ez, so we have selected out a class of preferred frames described by the quotient group O(3)/O(2).

It's essentially the same story for Cahill's reports of the data for Michelson-Morley type experiments whether with the traditional interferometers looking for a fringe shift when the interferometer is rotated, or, equivalently, looking for beats between two Helium-Neon lasers when their configuration is rotated (see the Catania paper), except now the group is more complicated. It is the 6-parameter Lorentz group O(1,3) that will spontaneously break in the boost sector I think to O(1,3)/O(3)! What does the boost piece of O(1,3) do? It connects inertial frames, in globally flat space-time. Let's forget general relativity at first. Obviously the local radii of curvature from general relativity will limit the size of the space-time domain - and Cahill takes pains to point out that he is talking about a local effect not to be confused with the dipole anomaly in the cosmic microwave background from motion of the Earth relative to large scale Hubble flow where H(t) = R(t)^-1dR(t)/dt in the FLRW metric - that's a different effect from Cahill's alleged one.

Just as in the ferromagnet, a particular ez is selected, so too here the empirical evidence is that some particular inertial frame normally equivalent under boosts in the vacuum is selected or preferred in a finite region of space-time. The local vacuum order parameter will be the class of equivalent tetrads (et,ex,ey,ez) that lives in the quotient order parameter space O(1,3)/O(3) analogous to the class of ferromagnetic triads that lives in the quotient order parameter space O(3)/O(2). Now if O(1,3)/O(3) has the same non-trivial topology as does O(3)/O(2) then we may get the Pioneer anomaly (hedgehog defect) as a bonus from the same theory? I am not sure of that as yet. It's a wild half-baked precognition at this point, which may be wrong or not even wrong.

Brush up your group theory, start doing it now.


Jack wrote:
The preferred orientation of the ferromagnetism in the ground state violation of O(3) is formally isomorphic to the preferred "rapidity" (i.e. the Wick-rotated orientation from Euclidean metric to hyperbolic metric) in the breakdown of O(1,3) in the physical vacuum. In ALL cases there is no violation of the dynamical symmetries. The action and the equations of motion are still tensor/spinor covariant under ALL the symmetries both spacetime and internal. This distinction between dynamical symmetry breakdown and spontaneous ground state breakdown was a struggle as P.W. Anderson chronicles in "A Career in Theoretical Physics" - even the great Eugene Wigner made his greatest blunder there back in the 60's I think on "electric charge superselection rules" violated in the BCS superconductor that is a macro-quantum coherent superposition of different numbers of bound electron pairs. This breaking of U(1) gauge symmetry is a "preferred frame" in the internal space, just as Cahill's et-al's absolute velocities give a "preferred frame" in ordinary space. The covariance of the fundamental laws of nature under all symmetry groups are NOT affected by this!

Z wrote:
It is not clear to me how this relates to the effects discussed by Cahill, which depend on the optical properties of the moving medium though which the MM light beam travels.

Jack wrote:
You are not understanding the key idea that you must see mathematically. You still do not get the analogy to the ferromagnet. What Cahill reports is exactly like a ferromagnet only the group G has changed from O(3) for the ferromagnet to O(1,3) for Cahill's reporting of the Michelson-Morley data, and also the Catania, Sicily group, they get a smaller number than Cahill.

You do not yet understand P.W. Anderson's "More is different", which is 2 in:

There are only two important ideas in theoretical physics

1. Local gauge invariance with compensating connection fields.

2. Spontaneous breaking of the symmetry of the vacuum leaving the dynamics intact.

That's Sarfatti's Theory of Everything.

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