Lecture 8
“The basic strategy of all modern theories of unification is to answer this question with the mechanism of:
Spontaneous symmetry breaking: The distinctions between the observed particles and interactions result from a vacuum state of the theory not being invariant under all the symmetries of the dynamics.”
“Comment 17: Associated with this is P.W. Anderson’s “More is different” – a general theory of the emergence of non-reductionistic qualitatively new orders of complex physical phenomena largely independent of the details of their substratum. Other related technical ideas are “ODLRO” Off-Diagonal-Long-Range-Order of reduced quantum density matrices that develop a macroscopic eigenvalue. When that happens the usual Born probability intepretation of micro-quantum theory breaks down for the “condensate” that obeys a LOCAL nonunitary nonlinear Landau-Ginzburg type equation for the macro-quantum coherent “order parameter”. There is still a linear unitary Schrodinger equation for the “normal” excitations out of the ground state at finite temperature as well as the virtual quanta inside the ground state even at zero temperature. The ground state condensate is the source and sink for these virtual quanta as well as real elementary excitations, both quasi-particles and collective modes (e.g. plasmons). Pandora’s Box of “spooky telepathic” acausal signal nonlocality opens up when “More is different” as explained by Antony Valentini now at The Perimeter Institute. Frank Wilczek shows how the ODLRO idea is generalized in the thin film anyons with fractional statistics in three exotic phases of the fractional quantum Hall effect, the anionic high temperature superconductor and the anyonic metal. Anyons are “charges” stuck to “flux tubes.” Are they the key to low power metric engineering of the fabric of spacetime, or is that simply a wrong idea in my mind? Time will tell. Still another related idea is Ken Wilson’s version of the “Renormalization Group” as well as “Wavelet Signal Transforms” and “Wigner Phase Space Densities.”
Back to Smolin:
This means that the properties that distinguish the different particles and forces from each other are due precisely to their relationship with a choice of background, which is a vacuum state of the theory. If a theory can have different vacuum states, which preserve different subgroups of the symmetries of the dynamics, then the properties of the particles and forces will differ in each.”
Comment 18: Yes, this is
V = G(Unbroken Unstable False Vacuum)/H(Broken Meta-Stable Vacuum)
V = Vacuum Manifold of energy degenerate local macro-quantum coherent order parameters.
The /\ zero point energy density in false vacuum > /\’ zero point energy density in the metastable vacuum of our pocket universe – for example.
The local order parameter is called the generic “Higgs Field”. There are several types of Higgs fields that split off at different scales.
Here we are in the coarse-grained cosmic landscape where /\ is truly a constant in space and time. We here ignore small-scale changes in /\ from additional torsion and conformal dilation fields that may well exist beyond Einstein’s 1915 theory of gravity as ONLY curvature.
Note everything Lenny Susskind does in Cosmic Landscape is only for large-scale coarse-graining washing out details smaller than 300 Megaparsecs (If I remember correctly?). That is, we can use the Cosmological Principle in this approximation in which coarse-grained 3D space is isotropic everywhere. Therefore it is also uniform. This is at a fixed global cosmic time in the Hubble flow assuming a very special Friedmann universe background with a timelike Killing isometry field. Then we can assume /\ is a parameter for that special kind of pocket universe. So this shows some of the assumptions of the cosmic landscape that Smolin is bringing to the fore.
The Vacuum Manifold V is for a fixed total Higgs field intensity held constant at the local minimum of the cosmic landscape (for a given set of maybe hundreds of 6D “torsion” Calabi-Yau internal parameters). What varies is a set of Goldstone phases. They are like phases of waves usually defined periodically mod 2pi as in the trigonometric functions. However, if they are imaginary then they are “boosts” along the infinite line as in the hyperbolic functions.
A topological defect is when the total Higgs field intensity drops down to zero so that the Goldstone phases are ill-defined. This is also called a Goldstone phase singularity. Note that in Lenny Susskind’s special case of the cosmic landscape that the Higgs intensity is CONSTANT in the entire universe at this large scale. This is very different from using the same idea on small scales where the Higgs intensity is a local variable field with several real components. We then use a fiber bundle. And the topological defects are localized in ordinary space. But in Lenny’s cosmic landscape a topological defect is a position in the cosmic landscape of hundreds of dimensions and is not at all in ordinary space where it is a uniform constant.
Note that in my theory
/\ = (Quantum of Area)^-1[1 - |Higgs Intensity|^2]
Where the Higgs Intensity is normalized as a pure number.
Therefore, /\ > 0 when Higgs intensity = 0 at a topological defect in the cosmic landscape where a single point describes an entire 3-geometry of a Friedman universe, including deSitter universes.
Note that the supersymmetric plateau where /\ = 0 is when the Higgs Intensity = 1. Note that /\ < 0 is possible since the macroquantum Higgs intensity is not at all a Born quantum probability measure but is a Bose-Einstein condensate of some kind.
The simplest case is a discrete set of two Goldstone phases 0 and Pi. V = G/H is then the unit circle S0 in zero dimensions, i.e. the two points +1 & -1. This is analogous to Fermi and Bose statistics in 3D systems. Note I used the word “analogous” since we are here talking of local macro-quantum order parameters not multi-particle quantum amplitudes where we exchange identical quanta so that the possibly NONLOCAL ENTANGLED quantum amplitudes fall into different irreducible “Young Pattern” representations of the discrete permutation group of order N! for N identical quanta. The topology idea here is “homotopy”. Two objects are homotopic if one can shape shift or morph into the other in a 1-1 reversible way without any tearing to avoid obstructions. In this case in 3D space, there are stable topological defects that are 2D “domain wall” boundaries. This is because the zero order homotopy group is non-trivial, i.e. larger than the identity element. Indeed it is essentially the cyclic group of order 2 under multiplication {1, -1} where -1x-1 = 1. What happens in 2D for anyons in thin films when V = S0?
The next case is V(S1) = G/H with only 1 Goldstone phase “Theta”. This is a unit circle in the plane. The stable topological defect in 3D space is then a string vortex, because the first homotopy group is equal to the integers. This corresponds to the original nonlocal electromagnetic Bohm-Aharonov effect of “flux without flux” as well as to quantized circulations in superfluid flow that is “vorticity without vorticity.” Surround the string defect with a closed loop. There is a closed non-exact Cartan 1-form A such that dA = 0 on the loop. One can think of dA =/= 0 concentrated on the string defect. Because the local Higgs field is single-valued, it follows that the “deRham” line integral of A round the closed loop is an integer times some quantum depending on the precise definition of A. Note that dA is a 2-form flux surface density that is the curl in ordinary vector calculus. This integer is the “winding number” and it has a screw sense. Its absolute value is the number of quanta of “flux without flux.” Therefore, a naïve application of Stoke’s theorem gives a “flux without flux” of
Or
This is the Bohm-Aharonov effect in the quantum case in which a physical shift of an electron wave interference pattern is observed ~ the distant magnetic flux. The uniform constant magnetic flux localized inside a tightly wound solenoid representing the string defect acts nonlocally at a distance on the electron that is always in a region of zero magnetic field B = dA = 0. An alternative explanation, that works naturally for superfluid flow is that the local vector potential has a direct physical effect even though it is not a gauge invariant quantity. The total global nonlocal integral giving the net fringe shift is gauge invariant however. No matter how you cut it. The spooky action at a distance actually observed violates classical common sense.
The next case in ordinary 3D space (not Lenny’s cosmic landscape case) is when
V = G/H = S2 with two Goldstone phases Theta & Phi. Here it is the second homotopy group that is non-trivial equal to the integers. This means there is a non-vanishing closed non-exact 2-form. It’s easy to construct this 2-form.
B = (dTheta)^(dPhi)
Where ^ is the anti-symmetric exterior multiplication of Cartan forms.
Theta & Phi are 0-forms.
dTheta & dPhi are non-parallel 1-forms. Indeed they are orthogonal on the unit 2-sphere S2 (latitude & longitude).
The topological defect is now a “monopole” point in 3-space (or a tiny ball of radius = coherence length).
Similarly the closed surface integral of B that surrounds at least one point defect where the Higgs field (order parameter) vanishes is an integer multiplied by some quantum. Now it’s Gauss’s theorem since
dB = 0 locally outside the point defect but in general d(2-form) is a divergence of a vector field in 3D space.
Therefore, we have “divergence without divergence” in which
No comments:
Post a Comment