Still about Equivalence Principle
Every time I have written about Equivalence Principle (briefly EP in the following ) in TGD framework I feel that it is finally fully and completely understood. But every time I am asked about EP and TGD, I feel uneasy and end up making just question "Could it be otherwise?". This experience repeated itself when Hamed made this question in previous posting.
Recall that EP in its Newtonian form states that gravitational and inertial masses are identical. Freely falling lift is the famous thought experiment leading to the vision of general relativity that gravitation is not a real force and in suitable local coordinate system can be eliminated locally. The more abstract statement is that particles falling freely in gravitational field move along geodesic lines. At the level of fields this leads Einstein's equations stating that energy momentum tensor is proportional Einstein tensor plus a possible cosmological term proportional to metric. Einstein's equations allow only the identification of gravitational and inertial energy momentum densities but do not allow to integrate these densities to four-momenta. Basically the problem is that translations are not symmetries anymore so that Noether theorem does not help. Hence it is very difficult to find a satisfactory definition of inertial and gravitational four-momenta. This difficulty was the basic motivation of TGD.
In TGD abstract gravitation for four-manifolds is replaced with sub-manifold gravity in M4× CP2 having also the symmetries of empty Minkowski space and one overcomes the mentioned problem. It is however far from clear whether one really obtains EP - even at long length scale limit! There are many questions in queue waiting for answer. What Equivalence Principle means in TGD? Just motion along geodesics in absence of non-gravitational forces or equivalence of gravitational and inertial masses? How to identify gravitational and inertial masses in TGD framework? Is it necessary to have both of them? Is gravitational mass something emerging only at the long leng scale limit of the theory? Does one obtain Einstein's equations or something more general at this limit - or perhaps quite generally?
What about quantum classical correspondence: are inertial and gravitational masses well-defined and non-tautologically identical at both quantum and classical level? Are quantal momenta (super conformal representations) and classical momenta (Noether charges for Kähler action) identical or does this apply at least to mass squared operators?
One can start from the fact that TGD is a generalization of string models and has generalization of super-conformal symmetries as symmetries. Quantal four-momentum is associated with quantum states - quantum superpositions of 3-surfaces in TGD framework. For the representations of super-conformal algebras (this includes both Virasoro and Kac-Moody type algebras) four-momentum appears automatically once one has Minkowski space and now one indeed has M4× CP2. One also obtains stringy mass formula. This happens also in TGD and p-adic thermodynamics and leads to excellent predictions for elementary particle masses and mass scales with minor input - one of them is five tensor factors in the representations of the super Virasoro algebra five tensor factors in the representations of the super Virasoro algebra. The details are more fuzzy since five tensor product factors for the Super Virasor algebra is the only constraint and it has turned possible to imagine many manners to satisfy the constraint. Here mathematician's helping hand would be extremely wellcome.
The basic question is obvious. Is there any need to identify both inertial and gravitational masses at the superconformal level? If so, can one achieve this?
- There are two super-conformal algebras involved. The supersymplectic algebra associated with the imbedding space (boundary of CD) could correspond to inertial four-momentum since it acts at the level of imbedding space and the super Kac-Moody algebra associated with light-like 3-surfaces to gravitational four momentum since its action is at space-time level.
- I have considered the possibility that the called coset representation for these algebras could lead to the identification of gravitational and inertial masses. Supersymplectic algebra can be said to contain the Kac-Moody algebra as sub-algebra since the isometries of light-cone boundary and CP2 can be imbedded as sub-algebra to the super-symplectic algebra. Could inertial and gravitational masses correspond to the four-momenta assignable to these two algebras? Coset representation would by definition identify inertial and gravitational super- conformal generators. In the case of scaling generator this would mean the identification of mass squared operators and in the case of they super counterparts identification of the four-momenta since the differences of super-conformal generators would annihilate physical states. The question whether one really obtains five tensor factors is far from trivial and here it is easy to fall to the sin of self deception.
A really cute feature of this approach is that p-adic thermodynamics for the vibrational part of either gravitational or inertial scaling generator does not mean breaking of super-conformal invariance since super conformal generators in the coset representation indeed annihilate the states although this is not the for the super-symplectic and super Kac-Moody representations separately. Note that quantum superposition of states with different values of mass squared and even energies makes sense in zero energy ontology.
- Second option would combine these two algebras to a larger algebra with common four-momentum identified as gravitational four-momentum. The fact that this four-momentum does not follow from a quantal version of Noether theorem suggests the interpretation as gravitational momentum. In this case the simplest manner to understand the five tensor factors of conformal algebra would be by assigning them to color group SU(3), electroweak group SU(2)L× U(1)(2 factors), symplectic groups of CP2 and light-cone boundary δ M4+.
While writing a response to the question of Hamed the following question popped up. Could it be that classical four-momentum assignable to Kähler action using Noether theorem defines inertial four-momentum equal to the gravitational four-momentum identified as four-momentum assignable to super-conformal representation for the latter option? Gravitational four-momentum would certainly correspond naturally to super-conformal algebra just as in string models. The identification of classical and quantal four-momenta might make sense since translations form an Abelian algebra,or more generally Cartan sub-algebra of product of Poincare and color groups. Even weaker identification would be identification of inertial and conformal mass squared and color Casimir operators. EP would reduce to quantum classical correspondence just and General Coordinate Invariance (GCI) would force classical theory as an exact part of quantum theory! This would be elegant and minimize the number of conjectures but could be of course be wrong.
One can argue that p-adic thermodynamics for the vibrational part of the total scaling generator (essentially mass squared or conformal weight defining it) breaks conformal invariance badly. This objection might actually kill this option.
Classical four-momentum and classical realization of EP
In the classical case the situation is actually more complex than in quantum situations due to the extreme non-linearity of Kähler action making also impossible naive canonical quantization even at the level of principle so that quantal counterparts of classical Noether charges do not exist. One can however argue that quantum classical correspondence applies in the case of Cartan algebra. Four-momenta and color quantum numbers indeed define one possible Cartan sub-algebra of isometries.
By Noether theorem Kähler action gives rise to inertial four-momentum as classical conserved charges assignable to translations of M4 (rather than space space-time surface). Classical four-momentum is always assignable to 3-surface and its components are in one-one correspondence with Minkowski coordinates. It can be regarded as M4 vector and thus also imbedding space vector.
Quantum classical correspondence requires that the Noetherian four-momentum equals to the conformal four-momentum. This irrespectively of whether EP reduces to quantum classical correspondence or not.
Einstein's equations have been however successful. This forces to ask whether classical field equations for preferred extremals could imply that the inertial four-momentum density defined by Kähler action is expressible as a superposition of terms corresponding to Einstein tensor and cosmological term or its generalization. If EP reduces to quantum classical correspondence, one could say that not only quantum physics but also quantum classical correspondence is represented at the level of sub-manifold geometry.
In fact, exactly the same argument that led Einstein to his equations applies now. Einstein argued that energy momentum tensor has a vanishing covariant divergence: Einstein's equations are the generic manner to satisfy this condition. Exactly the same condition can be posed in sub-manifold gravity.
- The condition that the energy momentum tensor associated with Kähler action has a vanishing covariant divergence is satisfied for known preferred extremals. Physically it states the vanishing of Lorentz 4-force associated with induced Kähler form defining a Maxwell field. The sum of electric and magnetic forces vanishes and the forces do not perform work. In static case the equations reduce to Beltrami equations stating that the rotor of magnetic field is parallel to the magnetic field. These equations are topologically highly interesting.
These conditions are satisfied if Einstein's equations with cosmological term hold true for the energy momentum tensor of Kähler action. The vanishing of trace of Kahler energy momentum tensor implies that curvature scalar is constant and expressible in terms of cosmological constant. If cosmological constant vanishes, curvature scalar vanishes (Reissner-Nordström metric is example of this situation and CP2 defines Euclidian metric with this property). Thus the preferred extremels would correspond to extremely restricted subset of abstract 4-geometries.
- A more general possibility - which can be considered only in sub-manifold gravity - is that cosmological term is replaced by a combination of several projection operators to space-time surface instead of metric alone. Coefficients could even depend on position but satisfying consistency conditions guaranteeing that the energy momentum tensor is divergenceless. For instance, covariantly constant projection operators with constant coefficients can be considered.
If this picture is correct (do not forget the objection from p-adic thermodynamcs), what remains is the question whether quantum classical correspondence is true for four-momenta or at least mass squared and color Casimir operators. Technically the situation would be the same for both interpretations of EP. Basically the question is whether inertial-gravitational equivalence and quantal-classical equivalence are one and same thing or not.
What is frustrating that the field equations of TGD are so incredibly non-linear that intuitive approach based on wild guesses and genuine thinking as opposed to blind application of calculational rules is the only working approach. I know quite well that for many colleagues intuitive thinking is the deadliest sin of all deadly sins. For them the ideal of theoretical physicist is a brainless monkey who has got the Rules. Certainly intuitive approach allows only to develop conjectures, which hopefully can be proven right or wrong and intuition can lead to wrong path unless it is accompanied by a critical attitude.
I am however an optimist. We know that it has been possible to develop perturbation theory for super symmetric version of Einstein action by using twistor Grassmann approach. Stringy variant of this approach with massless fermions as fundamental particles suggests itself in TGD. TGD Universe possesses huge symmetries and this should make also classical theory simple: I have indeed made proposal about how to construct general solutions of field equations in terms of what I call Hamilton-Jacobi structure. For these and many other reasons I continue to belief in the power of intuition.