### Universal formula for the hermitian square roots of density matrix

The construction of U-matrices discussed in previous posting led as a side product to a general formula for the M-matrices. Although the result is added to the previous posting, it deserves a separate discussion since it can be seen as a victory of the zero energy ontology providing zero energy states with Lie algebra structure and allowing to interpret hermitian square roots of density matrices as observables commuting with universal S-matrix. A generalization of Kac-Moody algebra emerges in which powers of S correspond to powers exp(i n φ).

Zero energy ontology replaces S-matrix with M-matrix and groups M-matrices to rows of U-matrix. S-matrix appears as factor in the decomposition of M-matrix to a product of hermitian square root of density matrix and unitary S-matrix interpreted in standard sense.

M_{i}=ρ^{1/2}_{i} .

Note that one cannot drop the S-matrix factor from M-matrix since M-matrix is neither unitary nor hermitian and the dropping of S would make it hermitian. The analog of the decomposition of M-matrix to the decomposition of Schrödinger amplitude to a product of its modulus and of phase is obvious.

The interpretation is in terms of square root of thermodynamics. This interpretation should give the analogs of the Feynman rules ordinary quantum theory producing unitary matrix when one has pure quantum states so that density matrix is projector in 1-D sub-space of state space (for hyper-finite factors of type II_{1} something more complex is required).

This is the case. M-matrices are in this case just the projections of S-matrix to 1-D subspaces defined by the rows of S-matrix. The state basis is naturally such that the positive energy states at the lower boundary of CD have well-defined quantum numbers and superposition of zero energy states does not contain different quantum numbers for the positive energy states. The state at the upper boundary of CD is the state resulting in the interaction of the particles of the initial state. Unitary of the resulting U-matrix reduces to that for S-matrix.

A more general situation allows square roots of density matrices which are diagonalizable hermitian matrices satisfying the orthogonality condition that the traces

Tr(ρ^{1/2}_{i}ρ^{1/2}_{j})=δ_{ij} .

The matrices span the Lie algebra of infinite-dimensional unitary group. The hermitian square roots of M-matrices would reduce to the Lie algebra of infinite-D unitary group. This does not hold true for zero energy states.

If one however assumes that S commutes with the algebra spanned by the square roots of density matrices and allows powers of S one obtains a larger algebra complely analogous to Kac-Moody algebra in the sense that powers of S takes the role of powers of exp(i nφ) in Kac-Moody algebra generators. The commutativity of S and density matrices means that the square roots of density matrices span symmetry algebra of S. The Hermitian sub-Lie-algebra commuting with S is large: for SU(N) it would correspond to SU(N-1)× U(1) so that the symmetry algebra is huge in infinite-D case.

A possible interpretation for the sub-space spanned by M-matrices proportional to S^{n} is in terms of the hierarchy of CDs. If one assumes that the size scales of CDs come as octaves 2^{m} of a fundamental scale then one would have m=n. Second possibility is that scales of CDs come as integer multiples of the CP_{2} scale: in this case the interpretation of n would be as this integer: this interpretation conforms with the intuitive picture about S as TGD counterpart of time evolution operator. This interpretation could also make sense for the M-matrice associated with the hierarchy of dark matter for which the scales of CDs indeed come as integers multiples of the basic scale.

If the square roots of density matrices are required to have only non-negative eigenvalues -as I have carelessly proposed in some contexts,- only projection operators are possible for Cartan algebra so that only pure states are possible. If one allows both signs one can have more interesting density matrices and this is the only manner to obtain square root of thermodynamics. Note that the standard representation for the Cartan algebra of finite-dimensional Lie group corresponds to non-pure state. For ρ=Id one obtains M=S defining the ordinary S-matrix. The orthogonality of this zero energy state with respect to other ones requires

Tr(ρ_{i}^{1/2})=0

stating that SU(N=∞) Lie algebra element is in question.

The reduction of the construction of U to that of S is an enormous simplification and reduces to the problem of finding the TGD counterpart of S-matrix. Note that the finiteness of the norm of SS^{†}=Id requires that hyper-finite factor of type II_{1} is in question with the definining property that the infinite-dimensional unit matrix has unit norm. This means that state function reduction is always possible only into an infinite-dimensional subspace only.

The natural guess is that the Lie algebra generated by zero energy states is just the generalization of the Yangian symmetry algebra of N=4 SUSY postulated to be a symmetry algebra of TGD (see this). The characteristic feature of the Yangian algebra is the multi-locality of its generators. The generators of the zero energy algebra are zero energy states and indeed form a hierarchy of multi-local objects defined by partonic 2-surfaces at upper and lower light-like boundaries of causal diamonds. Zero energy states themselves would define the symmetry algebra of the theory and the construction of quantum TGD also at the level of dynamics -not only quantum states in sense of positive energy ontology- would reduce to the construction of infinite-dimensional Lie-algebra! It is hard to imagine anything simpler!

For background see the chapter Construction of Quantum Theory: M-matrix.

## 8 Comments:

Interesting post. Indeed, a priori, there are no topological constraints preventing the use of CP^3, CP^4, etc. CP^2, however, is quite special when viewed as a complex subspace of OP^2, which has H*(OP^2;Z)~Z[a]/a^3.

Specifically, if H*(X;Z) is a polynomial algebra Z[a], truncated by a^n=0 with n > 3, then |a|=2 or |a|=4. In other words, we can for example construct CP^3, HP^3 and higher projective spaces over the complex and quaternion division algebras, but not over the octonions where OP^2 is maximal.

As OP^2 is the equivalence class of all rank 1 elements of J(3,O), embedding in infinite matrices to work with multiple copies of OP^2 seems out of the question, so one can choose to work with generalized Manin-Losev stacks instead, visualized as chains of projective planes, which are allowed to collide with each other.

I added the post to emphasize one important implication of zero energy ontology. The previous rather heavy posting provides a general solution to 32 year old challenge to construct S-matrix in TGD framework. The unification of thermodynamics and quantum theory by infinite-dimensional symmetry algebra identifiable as zero energy states is something incredibly beautiful.

The natural guess is that this algebra is actually the generalization of the Yangian symmetry of N=4 SUSY leading to a generalization of twistor program. The generators are zero energy states and indeed form a hierarchy of multilocal objects defined by partonic 2-surfaces at upper and lower light-like boundaries of causal diamonds. Yangian symmetry would not be only a symmetry of TGD but its generators would define the physical states and M-matrices! Could one consider more economic theory of everything!

I guess that you are talking from M-theory point of view in terms of branes. CP_2 follows in TGD framework from totally different premises: either from standard model spectrum or from number theoretical vision: M^4xCP_2 can be assigned to octonions and quaternions.

M-theorists would call space-time surfaces 4-branes in 8-D space but I prefer to talk just about 4-surfaces: there are no objects with other dimensions in TGD Universe. Space-time dimension D=4 is unique because it leads to extension of conformal symmetries (light-like 3-surfaces and light-like boundary of M^4 light-cone is metrically 2-D and possess infinite-D group of conformal symmetries and also isometries).

I was stating that there is topological justification for your use of CP^2 in TGD if it descends from an OP^2 construction. In any theory that makes use of the octonions, one cannot study scattering amplitudes in projective spaces higher than a plane, as such spaces do not exist.

Thank you. I see you point. One can however use octonions in many sense. I do not suggest octonionic quantum mechanics, not even quaternionic. Nor do I suggest quaternionic or octonionic generalization of holomorphy (I tried but failed as probably many others before me).

Rather, I suggest that octonions and quaternions at the level of geometry with spinor structure. Associativity becomes the basic dynamical principle classically.

What is unique for octonions that 8-D space allows octonionic representation of gamma matrices and therefore also Clifford algebra. Strictly speaking one cannot speak about "matrices" anymore.

The Minkowskian (Euclidian) regions of space-time surface are regarded as quaternionic or co-quaternionic in the sense that the algebra spanned by modified gamma matrices spans complexified quaternionic a (co-quaternionic) algebra as sub-algebra of complexified octonionic algebra. This means that it allows matrix representation. This is purely geometric condition and determines the classical dynamics of space-time surfaces. To avoid repeating X and co-X and complexified and so on let us simply say that space-time surfaces are quaternionic. This means that classical physics is associative and associativity is basic physical law classically.

One can go even further and identify string world sheets and partonic 2-surfaces which are essential part of TGD as hyper-complex and co-hypercomplex sub-manifolds of space-time surface and imbedding space. Braid strands could be perhaps identified as real or co-real sub-manifolds. This would look very nice but a precise characterization of what one means with this and of course also real proof is lacking.

Introducing localized complexified octonionic Clifford algebra for the imbedding space one can say that that space-time surfaces correspond local sub-algebras with gamma matrix valued functions restricted on quaternionic space-time surfaces.

Yes, the complexified octonion algebra, i.e., the bioctonion algebra, is very interesting from many different perspectives. The octonions and split octonions are subalgebras of the bioctonions. It is well known that 2x2 Hermitian matrices over the octonions and split-octonions can be used to represent vectors in (9,1) and (5,5) signature spacetimes.

In D=6, N=8 supergravity, the (5,5) signature arises while studying the charge vectors for black strings. The lightlike vectors, which are rank one with zero determinant, correspond to 1/2 BPS black strings.

Using a non-compact Hopf fibration, we can map two component split octonion spinors to the space of 2x2 rank one matrices via the hyperboloid map H^{8,7}->H^{4,4} with fiber H^{4,3}. This is a generalized version of the map used by Arkani-Hamed in studying twistor scattering amplitudes, where instead of mapping C^2 spinors to CP^1 we are mapping to the hyperboloid H^{4,4}, the space of lightlike 1/2 BPS black hole charge vectors.

Sorry for typo: D=8-->D+2=10 was of course my intention.

Thanks for the clarification, Matti. So is TGD ultimately a theory where the basic objects are microscopic wormholes? If so, what is the precise mathematical representation of such wormholes and how does one go about recovering the standard model spectrum in this framework?

One can say so but it must be made clear that wormholes are not quite the same as in general relativity.

Space-time regions with both Euclidian and Minkowskian signature of induced metric are possible for space-time surfaces. The regions of space-like signature are identified as counterparts of Feynman graphs with a deformation of CP_2 vacuum extremal representing the particle line.

Regions with Minkowskian signature correspond to space-time in the usual sense.

There is again a duality involved: the predictions of the theory can be formulated either in terms of Kaehler action whose value for preferred extremal defines Kahler function of WCW ("world of classical worlds") restricted to Euclidian or Minkowskian regions. Wick rotation is the analog in QFT.

Wormhole throat is the 3-D light like surface at which the signature of the induced metric changes. This can occur also for single space-time sheet in which case one does have only throat. Particle quantum numbers are associated with the wormhole throats.

The deformation of CP_2 type vacuum extremals with Euclidian signature of metric are also associated with the touching of two parallel Minkowskian space-time sheets and in this case one has two wormhole throats. For instance, bosons corresponds to this kind of objects basically.

The size of smallest wormhole throats is about CP_2 size scale: about 10^4 Planck lengths so that these wormhole throats are only topologically similar to those of GRT.

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