The challenge of a theoretician working with higher dimensions is to explain why our world seems to be 4-dimensional. No explanation for this fact has emerged in super string models: Kaluza-Klein approach led to Calabi-Yau's but led to the landscape catastrophe. Also 3-dimensional branes have been proposed to be in special role. The recent "solution" of the problem is actually rewarming of an old Kaluza-Klein spirited proposal. The idea is that somehow the symmetry-group of 10-dimensional space would reduce to SO(3) and this could be interpreted as isometries of 3-D Euclidian space. What the Japanese group has done are lattice calculations for what they call Matrix model, which is one of the extremely fuzzy variants of M-theory. It is suggested to provide nothing less than a non-perturbative formulation of M-theory.
In TGD framework the explanation for both the dimension 8 of imbedding space and the dimension D=4 of its Minkowski space factor and the dimension D=4 of space-time surface images from symmetry considerations. The world of classical worlds (space-time surfaces identified as preferred extremals of Kähler action) has Kähler geometry only if it has infinite-D isometry group. The conformal symmetries of the light-like 3-surfaces defining the surfaces at which the induce metric changes its signature and at the same time also generalized Feynman graphs extend the ordinary 2-D conformal symmetries and one obtains infinite-D isometr group. This is possible only for 4-D space-time surfaces. Also the light-cone boundary of light-cone of 4-D Minkowski space has extended conformal symmetries so that 4-D Minkowski space is unique.
There are also other reasons- in particular those related to the dimensions of classical number fields selecting these dimensions and M4× CP2. It is of course useless to tell this to the colleagues: they refuse to listen to a pariah and prefer to contine beating their intelligent heads on the wall. The weird activities with Matrix represent only one example of this acticity.
The derivation of Matrix model
I decided to check what Matrix model means -if anything- and found an article by Tom Banks and some others about Matrix model. It is Christmas Eve so that I want to put it mildly: my feelings were mixed. There are long sequences of hand waving arguments before one ends up with what is christened as Matrix model.
- One begins with M-theory and considers branes, which as such are very questionable objects physically. One suddenly decides that 0-branes representing point-like particles are fundamental rather than strings which would be analogous to 1-branes. All higher dimensional branes would be constructed from 0-branes. This would mean return to old days when particles where point-like. But do let this to disturb you. It is good to change the liturgy sometimes.
- After this one puts an infinite number of 0-branes- point like particles- on top of each other at same point of 10-D space-time. Like putting infinite-number of point like particles on top of each other. Stay calm and trust the M-theoretician. He knows. Then M-theoretician tells that AdS-correspondence says that the 10 coordinates assignable to the the points form actually matrices in SU(N)×U(1) at the limit of infinite N so that the 10 coordinates have become 10 non-commutating matrix valued coordinates. If you have difficulties swallowing this remember that everyone learns that you are an imbecille with an intelligence quotient of amoeba if you say this aloud.
- The AdS/CFT approach relies on the physical idea that the particles at different branes defined U(1) factors of gauge group. If the branes in the stack carry fermionic single particle states states one can indeed construct from them generators of SU(N)×U(1). This idea does not look non-sense to me: for finite number of N it might indeed give rise to a dynamical gauge group having interpretation as gauge symmetry allowing to describe finite measurement resolution in TGD framework. This however gives rise to gauge potentials with non-commutative gauge group, not non-commutative coordinates of 10-D space-time. This is a little problem- actually not so little - but let us trust M-theoreticians and identify vector potentials with 10-D space-time coordinates.
- M-theoretician decides that the dynamics of everything is dictated by Yang-Mills Hamiltonian restricted to the world line of the stack of 0-branes. It should be possible to replace Yang-Mills Hamiltonian with corresponding Lagrangian and I shall indeed do this in the sequel to see what results.
Classical Lagrangian Matrix model
Just for fun I decided to check what YM Lagrangian in 1-D time-line would give.
- The field equations express the vanishing of non-Abelian gauge currents. Gauge condition is naturally A0=0 corresponding to vanishing of the time component of gauge potential. There would be no time in this Universe but let us not bet worried. One can write YM equations and they state the vanishing of YM currents: Jα=0.
- J0=0 can be solved exactly and the solutions says that the other components of the potential are like the coordinates of a free particles Ai= xi(0)+vi(0) t. Particle interpretation happens to make sense except that time is constant.
- Classically the conditions Ji=[Aj,[Ai,Aj]]=0 give for matrix conditions on initial values of xi(0) and vi(0). Four conditions for each component corresponding to powers tn, n=0,1,2,3. These equations are symmetric under the exchange of positions and momenta- kind of duality. Classically they represent infinite number of polynomial equations of third order in variables xi(0) and pi(0) restricting the motion to an algebraic surface in infinite-D space of 9-vectors with matrix valued components. The polynomials are homogenous so that scalings of initial values are symmetries.
- An interesting ansatz consist of restricting the vector potentials to subgroup of U(∞). One obtains infinite hierarchy of solutions for which the motion of pseudo particle takes place in finite-dimensional subgroup U(n) or its subgroup. As a special case on also obtains solutions which correspond to Cartesian products ∏ U(ni). The interpretation as N-particle solutions is attractive. For Cartesian powers of U(1):s the constraints are identically satisfied. For Cartesian powers of SU(2) one obtains what looks like particles in 3-space with coefficients of Lie-algebra generators of SO(3) interpreted as coordinates of 3-space.
Quantum variant of Lagrangian Matrix model
The classical solvability apart from constraints suggests that also the quantized variant of the theory is solvable.
- Let us assume that wave functions are in the space defined by initial values xi(0). The standard commutation relations for the matrix elements of Ai and pi= dAi/dt= pi(0) reduce to those for xi(0) and pi(0). The matrix elements of these matrices are just like the position and momentum coordinates for free particle in Euclidian space. The conditions Ji=0 reduces to four sets of equations corresponding to powers n=0,1,2,3 of t. n=0 gives infinite number of third order polynomial equations for initial values of xi(0). n=1 gives second order partial differential equations linear in xi(0). n=2 gives gives first order partial differential equations quadratic in xi(0). n=3 gives infinite numer of third order partial differential equations in pi(0).
- Also in the quantum one has the hierarchy of solutions assignable to subgroups of U(∞) and their Cartesian products. If one wants something like emergent 3-space I would assign it to wave functions corresponding to the Cartesian powers of SU(2). I would try to show that the constraints favor this kind of solutions and that for higher dimensions of SU(N) or its subgroups the number of conditions is too large. I would also ask whether the fact that SU(2) Lie-algebra is in some sense the building brick of higher dimensional Lie algebras as the general form of the presentation of Lie algebras demonstrates could somehow mean that also more complex solutions are perturbations of those for Cartesian powers of SU(2). These questions are useful but it turns out that another 3-D Lie algebra- Heisenberg algebra- seems to be more interesting as effective gauge algebra.
The case of SU(2)
Are the resulting quantum equations sensible? For simplicity one can consider solutions restricted to SU(2) subgroup in the sense that the subgroup of U(∞) is N:th Cartesian power of SU(2) with interpretation in terms of N-particle state.
- There are 3× 9=27 coordinates xi(0). There are 3× 9 equations for n=0. The naive conclusion consistent with the scaling invariance and counting the number of conditions is that the only solution is xi(0)=0.
- This conclusion is too hasty. One can have solutions for which xi is restricted to 1-D subspace of E9. In this case the conditions expressible in terms of 3-D cross product are identically satisfied since cross products vanish. In this case the situation is indeed 3-dimensional and one can say that solution describes a pointlike free particle moving in 3-D space defined by the SU(2) Lie algebra valued preferred coordinate of E9. I have failed to to find more general solutions to the conditions.
- One can also consider wave functions restricted to 1-dimensional sub-manifolds of E9. Constraint conditions are identifically satisfied since the operators corresponding to n=1,2,3 vanish identically. The dimension of the higher dimensional space place no role since everything depends on the 3-D cross product characterizing quaternions and making SU(2) as a gauge group unique. The overall wave function would be restricted to a 2-D plane M2 of M10. Amusingly, in TGD framework preferred planes M2 of M4 play a key role.
Effectively 3-dimensional solution ansatz based on Heisenberg algebra
One obtains effectively 3-dimensional solution ansatz by restricting the consideration to 3-D subspace of E9.
- Using the notation [xi(0) =Ai the solution ansatz reads is of form
(A1, A2, t[A1,A2])
Here t is arbitrary real parameter. n=0 contributions to the gauge currents vanish if one has
This implies that the algebra generated by Ai is a matrix representation of Heisenberg algebra and the Lie algebra of gauge group is replaced by non-compact Heisenberg group. Finite-dimensional representations of this algebra are non-unitary but infinite-dimensional representations can be unitary.
- For n=3 the equations are also identically satisfied by the Heisenberg algebra property.
- The equations for n=1 and n=2 one obtains conditions on the wave function. Heisenberg algebra characterizes harmonic oscillator and for the vacuum state the condition that annihilation operator creates vacuum can be regarded as the analog of this conditions. On physical grounds one expects something like harmonic oscillator wave functions. Could one apply the analog of standard quantization meaning that the wave functions depend only on single coordinate -say that associated with x1(0).
A possibility that comes into mind is that action of the differential operators representing momentum p1(0) is given by
p1(0)Ψ= kx1(0) Ψ
so that the conditions on Ψ reduce to those for n=0 and are identically satisfied by Heisenberg algebra property.
This kind of condition is also analogous to a covariant constancy condition and makes sense for single coordinate. The value of the parameter k characterizes the width harmonic oscillator Gaussian (scale of energy spectrum for harmonic oscillator).
What is interesting is that one would have something analogous to what happens in TGD. The solution would be 3-dimensional and its time evolution would span a 4-D subspace. Uncertainy Principle for Heisenberg algebra would allow the wave function depend on single coordinate and one would effectively have 2-dimensional plane.
- Whether it makes sense to generalize the solution ansatz so that the linear 3-D subspaces of E9 are replaced with more general 3-D sub-manifolds is not clear.
A more general solution ansatz is obtained by considering the Cartesian powers of 3-D Heisenberg algebra. One obtains also solutions for which Qi= Ai, i=1,...,4 and Pi=Ai, i=5,...,8 commute the identity matrix in the generalization of Heisenberg algebra generated by the phase space coordinates Qi and Pi. Again unitary representations are infinite-dimensional. In this case the harmonic oscillator wave function would be effectively 4-dimensional.
Could N=∞ limit for gauge theory interpreted in terms of generalized Heisenberg algebra
Could one think of D=4 gauge theory at the limit when the dimension N of gauge group approaches infinity. Just for Christmas fun let us assume that it is literally infinite and apply the idea that the gauge potentials form a 3-D Heisenberg algebra.
- Assume A0=0 as a gauge condition. Assume also Heisenberg algebra
[A1,A2]= m A3
with other commutators vanishing. This condition makes sense for Hermitian fields strength in an infinite-D representation of Heisenberg algebra using infinite-D unitary gauge group. The new element is the interpretation of gauge potentials as generators of local Lie-algebra representable as Hermitian objects in infinite-D unitary gauge algebra.
- Consider first the vanishing of J0. The expression of the electric field strength reduce by gauge condition to F0i = -∂tAi and is identical to that of Abelian gauge field theory.
- What about the equations stating the vanishing of Ji? In this case the Heisenberg algebra guarantees the vanishing of the trilinear terms in the currents. The gauge field strengths F13 and F23 reduce also to their Abelian expressions. F12= ∂2A1 -∂1A2+mA3. Field equations stating the vanishing of J3 are Maxwellian. In J1=0 ( J2=0) the term m∂2A3 (-m∂1A3) appears as effective source term so that the field serves as a linear sources for itself.
- Again one might apply the analogy with harmonic oscillator and argue that the wave functions in the space of field configurations can depend only on single component of gauge potential- say A1. The interpretation would be in terms of polarization direction. The special role of A3 corresponds to the direction of 3-momentum in gauge theory.
- Also now Cartesian powers of Heisenberg algebra are possible and one obtains the analogs of many-particle states with varying directions of polarizations. One can assign to the solution also a preferred plane of M2 defined by time axes and the preferred component of gauge potential. With a good will one might say that M2 is analogous to string world sheet. One can ask whether symplectic transformations in E2 could give rise to more general solutions with a local polarizaion direction.
What would be non-trivial is that the field equations would linearize completely by using Heisenberg algebra ansatz and the otherwise difficult-to-treat local non-linearity would reduce to well-understod local Lie algebra commutations.
What can one conclude from this exercise?
- A rather general solution ansatz is expressible as a product solution corresponding to N:th Cartesian power of U(1). For U(1) factors the solutions can be said to be 9-dimensional since there are no constraints on the 9 coordinates xi(0). Another solution -even more trivial - solution ansatz reduces to 1-dimensional sub-space of E9 and has vanishing gauge field strength for any gauge algebra. Constraint equations are trivially satisfied.
- A genuinely 3-dimensional solution ansatz is obtained if one allows the gauge potentials to generate 3-D Heisenberg algebra. If the matrix algebra is infinite-dimensional, sub-algebra of U(&infty;), the matrices in question can be made unitary. Solution ansatz seems to work if one performs standard quantization in Heisenberg algebra so that wave functions depend on single coordinate xi only. This solution ansatz generalizes by replacing the Heisenberg group with its Cartesian power.
- The idea that 3-D space somehow emerges from Matrix theory is based on the reduction of isotropy group from SO(9) ⊂ SO(1,9) to SO(3). One might perhaps say that for the Heisenberg ansatz 3-dimensionality is obtained in this sense. Heisenberg group acts as gauge group. When the Cartesian power of Heisenberg group, the solution could be seen as describing free particles moving in 3-D space.
- The Lagrangian variant of the Matrix model is a nice toy model but to my opinion has very little to do with physics.