In the article Maxwellian electrodynamics is taken as a simpler toy example.
- One can assign to gauge transformations conserved charges. Gauge invariance tells that these charges vanish for all gauge transformations, which approach trivial transformation at infinity. Now however it is assumed that this need not happen. The assumption that action is invariant under these gauge transformations requires that the radial derivative of the function Φ defining gauge transformation approaches zero at infinity but gauge transformation can be non-trivial in the angle coordinates of sphere S2 at infinity. The allowance of these gauge transformations implies infinite number of conserved charges and QED is modified. The conserved gauge charges are generalizations of ordinary electric charged defined as electric fluxes (defining zero energy photons too) and reduce to electric gauge fluxes with electric field multiplied by Φ.
- For Maxwell's theory the ordinary electric charged defined as gauge flux must vanish. The coupling to say spinor fields changes the situation and due to the coupling the charge as flux is expressible in terms of fermionic oscillator operators and those of U(1) gauge field . For non-constant gauge transformations the charges are at least formally non-trivial even in absence of the coupling to fermions and linear in quantized U(1) gauge field.
- Since these charges are constants of motion and linear in bosonic oscillator operators, they create or annihilate gauge bosons states with vanishing energy: hence the term soft hair. Holographists would certainly be happy since the charges could be interpreted as representing pure information. If one considers only the part of charge involving annhilation operators one can consider the possibility that in quantum theory physical states are eigenstates of these "half charges" and thus coherent states which are the quantum analogs of classical states. Infinite vacuum degeneracy would be obtained since one would have infinite number of coherent states labelled by the values of the annihilation operator parts of the charges. A situation analogous to conformal invariance in string models is obtained if all these operators either annihilate the vacuum state or create zero energy state.
- If these U(1) gauge charges create new ground states they could carry information about matter falling into blackhole. Particle physicist might protest this assumption but one cannot exclude it. It would mean generalization of gauge invariance to allow gauge symmetries of the proposed kind. What distinguishes U(1) gauge symmetry from
non-Abelian one is that fluxes are well-defined in this case.
- In the gravitational case the conformal transformations of the sphere at infinity replace U(1) gauge transformations. Usually conformal invariance would requite that almost all conformal charges vanish but now one would not assume this. Now physical states would be eigentates of annihilation operator parts of Virasoro generators Ln and analogous to coherent states and code for information about the ground state. In 4-D context interpretation as strong form of holography would make sense.The critical question is why should one give up conformal invariance as gauge symmetry in the case of blackholes.
- In TGD framework conformal transformations of boundary of CD correspond to the analogs of BMS transformations. Actually conformal transformations of not only sphere (with constant value of radial coordinate labeling points of light rays emerging from the tip of the light-cone boundary) but also in radial degrees of freedom so that conformal symmetries generalize. This happens only in case of 4-D Minkowski space and also for the light-like 3-surfaces defining the orbits of partonic 2-surfaces. One actually obtains a huge generalization of conformal symmetries. As a matter of fact, Bee wondered whether the information related to radial degrees of freedom is lost: one might argue that holography eliminates them.
- Amusingly, one obtains also the analogs of U(1) gauge transformations in TGD! In TGD framework symplectic transformations of light-cone boundary times CP2 act like U(1) gauge transformations but are not gauge symmetries for Kähler action except for vacuum extremals! This is assumed in the argument of the article to give blackhole its soft hair but without any reasonable justification. One can assign with these symmetries infinite number of non-trivial conserved charges: super-symplectic algebra plays a fundamental role in the construction of the geometry of "World of Classical Worlds" (WCW).
At imbedding space level the counterpart for the sphere at infinity in TGD with the sphere at which the lightcone-boundaries defining the boundary of causal diamond (CD) intersect. At the level of space-time surfaces the light-like orbits of partonic 2-surfaces at which the signature of the induced metric changes are the natural counterparts of the 3-surface at infinity.
In TGD framework Noether charges vanish for some subalgebra of the entire algebra isomorphic to it and one obtains a hierarchy of quantum states (infinite number of hierarchies actually) labelled by an integer identifiable in terms of Planck constant heff/h=n. If colleagues managed to realize that BMS has a huge generalization in the situation when space-times are surface in in M4×CP2, floodgates would be open.
One obtains a hierarchy of breakings of superconformal invariance, which for some reason has remained un-discovered by string theorists. The natural next discovery would be that one indeed obtains this kind of hierarchy by demanding that conformal gauge charges still vanish for a sub-algebra isomorphic with the original one. Interesting to see who will make the discovery. String theorists have failed to realize also the completely unique aspects of generalized conformal invariance at 3-D light-cone boundary raising dimension D=4 to a completely unique role. To say nothing about the fact that M4 and CP2 are twistorially completely unique. I would continue the list but it seems that the emergence super string elite has made independent thinking impossible, or at least the communications of the outcomes of independent thinking.
- The first thing to notice is that Kähler gauge potentials are not the primary dynamical variables. This role is taken by the imbedding space coordinates. The symplectic transformations of CP2 act like gauge transformations mathematically but affect the induced metric so that Kähler action does not remain invariant. The breaking is small due to the weakness of the classical gravitation. Indeed, if symplectic transformations are to define isometries of WCW, they cannot leave Kähler action invariant since the Kähler metric would be trivial! One can deduce symplectic charges as Noether charges and they might serve as analogs fo the somewhat questionable generalized gauge charges in HPS proposal.
- If the counterparts of the gauge fluxes make sense they must be associated with partonic 2-surfaces serving as basic building bricks of elementary particles. Field equations do not follow from independent variations of Kähler gauge potential but from that of imbedding space coordinates. Hence identically conserved Kähler current does not vanish for all extremals. Indeed, so called massless extremals ( MEs) can carry a non-vanishing light-like Kähler current, whose direction in the general case varies. MEs are analogous to laser beams and if the current is Kähler charged it means that one has massless charged particle.
- Since Kähler action is invariant also under ordinary gauge transformations one can formally derive the analog of conserved gauge charge for non-constant gauge transformation Φ. The question is whether this current has any physical meaning.
One obtains current as contraction of Kähler form and gradient of Φ:
jαΦ= Jαβ∂βΦ ,
which is conserved only if Kähler current vanishes so that Maxwell's equations are true or if the contraction of Kähler current with gradient of Φ vanishes:
jαΦ∂αΦ=0 .
The construction of preferred extremals leads to the proposal that the flow lines of Kähler current are integrable in the sense that one can assign a global coordinate Ψ with them. This means that Kähler current is
proportional to gradient of Ψ:
jαΦ= gαβ ∂βΨ .
This implies that the gradients of Φ and Ψ are orthogonal. If Kähler current is light-like as it is for the known extremals, Φ is superposition of light-like gradient of Ψ and of two gradients in a sub-space of tangent space analogous to space of two physical polarizations. Essentially the local variant of the polarization-wave vector geometry of the modes of radiative solutions of Maxwell's equations is obtained. What is however important that superposition is possible only for modes with the same local direction of wave vector (∇Ψ) and local polarization.
Kähler current would be scalar function k times gradient of Ψ :
jαΦ= kgαβ∂βΨ .
The proposal for preferred extremals generalizing at least MEs leads to the proposal that the extremals define two light-like coordinates and two transversal coordinates.
- The conserved current decomposes to a sum of interior and boundary terms. Consider first the boundary term. The boundary contributions to the generalized gauge charge is given by the generalized fluxes
Qδ,Φ= ∮ JtnΦ g1/2
over partonic 2-surfaces at which the signature of the induced metric changes from Euclidian to Minkowskian. These contributions come from both sides of partonic 2-surface corresponding to Euclidian and Minkowskian metric and they differ by a imaginary unit coming from g1/2 at the Minkoskian side. Qδ,Φ could vanish since g1/2 approaches zero because the signature of the induced metric changes at the orbit of the partonic 2-surfaces. What happens depends on how singular the electric component of gauge potential is allow to be. Weak form of electric magnetic duality proposed as boundary condition implies that the electric flux reduces to magnetic flux in which case the result would be magnetic flux weighted by Φ.
- Besides this there is interior contribution, which is Kähler current multiplied by -Φ:
Qint,Φ= ∫ jtΦ g1/2 .
This contribution is present for MEs.
- Could one interpret these charges as genuine Noether charges? Maybe! The charges seem to have physical meaning and they depend on extremals. The functions Φ could even have some natural physical interpretation. The modes of the induced spinor fields are localized at string world sheets by strong form of holography and by the condition that electric charge is well defined notion for them. The modes correspond to complex scalar functions analogous to powers zn associated with the modes of conformal fields. Maybe the scalar functions could be assigned to the second quantized fermions. Note that one cannot interpret these contributions in terms of oscillator operators since the second quantization of the induced gauge fields does not make sense. This would conform with strong form of holography which in TGD framework sense that the descriptions in terms of fundamental fermions and in terms of classical dynamics of Kähler action are dual. This duality suggest that the quantal variants of generalized Kähler charges are expressible in terms of fermionic oscillator operators generating also bosonic states as analogs of bound states. The generalized charge eigenstates might be also seen as analogs of coherent states.
For a summary of earlier postings see Links to the latest progress in TGD.
4 comments:
This paper is on the topic... can you comment from TGD's perspective?
"
The singularity of the Cayley transform implies that some energy levels, usually associated with edge states, acquire an infinity energy when by an adiabatic change the boundary cond
itions reaches the Cayley submanifold C−. In this sense topological transitions require an infinite a mount of quantum energy to occur, although the description of the topolog
ical transition in the space M is smooth. This fact has relevant implications in string the
ory for possible scenarios with joint descriptions of open and closed strings. In the partic
ular case of elliptic self–adjointboundary conditions, the space C −can be identified with a Lagrangian submanifold ofthe infinite dimensional Grassmannian. The corresponding C
ayley manifold C− is dual of the Maslov class of M . The phenomena are illustrated with some simple low dimensional examples"
Cheers,
Stephen
Sorry, missing link in previous comment, here is the URL
http://arxiv.org/abs/hep-th/0403048
" Global Theory of Quantum Boundary Conditions and Topology Change"
this is related to that orthogonal basis whose elucidation would would allow one to construct the proof of the Riemann hypothesis in an elegant way
--Stephen
Pleasant surprise: I had the feeling that the article is readable by me. It was about "a global theory of boundary conditions" and Cayley space is the space of general boundary conditions. At least I had the experience of understanding what they are talking about!
Consider first what boundary conditions mean in TGD.
a) The boundary conditions at the ends of causal diamond (and at the light-like orbits of partonic 2-surfaces) define what preferred extremal property means and realize strong form of holography: 3-D system behaves almost like 2-D one so that only the data at partonic 2-surfaces and string world sheets (this implies "almost") matters.
b) The conditions state that the Noether charges of sub-algebras of various generalised conformal algebras involved (in particular, symplectic algebra of light-cone boundary times CP_2) vanish. This algebra is isomorphic to the entire algebra so that one has fractal hierarchies of isomorphic sub-algebras. By including fermion sector one obtains super-symplectic algebra realised in terms of second quantized fermions.
This is roughly the physical picture. The Cayley space of general boundary conditions is quite too large in TGD framework. One must restrict it to the above boundary conditions.
a) The fractal hierarchy of sub-algebras of conformal algebras isomorphic to it - perhaps symplectic algebra is enough - would define a hierarchy of linear spaces of boundary conditions. This kind of algebra is identifiable as the tangent space of infinite-D "conformal group" (includes Kac Moody type groups and supersymplectic group) defining the non-linear structure. The generators O_m , m=k*n annhilate the physical states, n characterizes the subalgebra and would correponds to Planck constant h_eff/h=n. The number of physical degrees of freedom would be still infinite.
b) Finite measurement resolution suggests stronger boundary conditions. If the conditions are strengthened so that not only elements with conformal weights coming as multiples k*n of n annihilate or produce zero energy states from physical states but also also the elements for which m>=n do this, the space of physical degrees of freedom becomes finite-D. For n=1 one obtains ordinary conformal gauge conditions.
c) Lie-algebraically this means that the commutator algebra of sub algebra with the full Lie algebra annhilates the physical states and sub-group behaves effectively like normal subgroup so that the coset space of full conformal group with the subgroup is a finite-D group characterizing the physical degrees of freedom. One would perhaps obtain a dynamical gauge symmetry: finite-dimensional ADE type Lie groups assignable to the hierarchy of Jones inclusions would appear as dynamical gauge groups the situation at given hierarchy level. One might say that TGD is able to mimick any ADE type gauge theory dynamically: electrowek and and color symmetires are of course different thing. This mimicry would happen at the level of dark matter.
Gauge conditions would effectively reduce the number of superconformal degrees of freedom to finite number. Hawking et al recent paper suggests weakening of conformal gauge conditions. Thinking five minutes this leads to the hierarchy of conformal algebras ( http://matpitka.blogspot.fi/2016/01/about-new-proposal-of-hawking-perry-and.htm ).
Authors talk about topology of solutions. Topology of space-time surfaces would be central also now.
h_eff/h=n labelling given level of fractal hierarchy of sub-algebras would correspond to n-fold coverings of space-time sheet with sheets co-inciding at the boundaries of CD. The sheets differing by a gauge transformation generated by algebra with weights larger than n would be regarded as equivalent so that light-cone orbits of partonic 2-surface would define gauge equivalence classes as in gauge field theories.
This condition conforms with the view about quantum criticality. Quantum criticality implies quantum fluctuations and loss of complete predictability. Now non-determinism would correspond to n different orbits connecting initial and final 3-surfaces. Long range fluctuations associated with criticality would correspond to increase of h_eff scaling up scale of quantum coherence. Phase transition increasing n_1 to n_2 would increase the number of sheets of the covering.
One might perhaps interpret the situation also differently: replace 3-surfaces unions of space-like 3-surface at both ends of CD and of light-like partonic orbits to get closed 3-surfaces which are everywhere space-like or light-like (nowhere timeline). Quantum measurement in ZEO theory however favours the original interpretation.
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