- You start by assuming that a tachyon having negative mass squared: m(ν)2<0 and assume that super-luminal velocity is in question. The point is that you know the value of the superluminal velocity v(1+ε)c, ε≈ 10-5. You can calculate the energy of the neutrino as
E= |m(ν)|[-1+ v2/(v2-1)]1/2,
|m(ν)|=(-m(ν)2)1/2 is the absolute value of formally imaginary neutrino mass.
- In good approximation you can write
E= |m(ν)|[-1+ (2ε-1/2]1/2 ≈|m(ν)| (2ε)-1/2.
The order of magnitude of |m(ν)| is not far from one eV - this irrespective of whether neutrino is tachyonic or not. Therefore the energy of neutrino is very small: not larger than keV. This is in a grave contradiction whith what is known: the energy is measured using GeV as a natural unit so that there is discrepancy of 6 orders of magnitude at least. One can also apply energy conservation to the decay of pion to muon and neutrino and this implies that muon has gigantic energy: another contradiction.
What is amusing that this simple kinematic fact was not noticed from beginning. In any case, this finding kills all tachyonic models of neutrino super-luminality assuming energy conservation, and gives additional support for the TGD based explanation in terms of maximal signal velocity, which depends on pair of points of space-time sheet connected by signal and space-time sheet itself characterizing also particular kind of particle.
Even better, one can understand also the jitter in the spectrum of the arrival times which has width of about 50 ns in terms of an effect caused fluctuations in gravitational fields to the maximal signal velocity expressible in terms of the induced metric. The jitter could have interpretation in terms of gravitational waves inducing fluctuation of the maximal signal velocity c#, which in static approximation equals to c#=c(1+Φgr)1/2, where Φgr is gravitational potential.
Suprisingly, effectively super-luminal neutrinos would make possible gravitational wave detector! The gravitational waves would be however gravitational waves in TGD sense having fractal structure since they would correspond to bursts of gravitons resulting from the decays of large hbar gravitons emitted primarily rather than to a continuous flow (see this). The ordinary detection criteria very probably exclude this kind of bursts as noise. The measurements of Witte attempting to detect absolute motion indeed observed this kind of motion identifiable as a motion of Earth with respect to the rest frame of galaxy but superposed with fractal fluctuations proposed to have interpretation in terms of gravitational turbulence - gravitational waves.
For details see the earlier posting, the little article Could the measurements trying to detect absolute motion of Earth allow to test sub-manifold gravity? or the chapter TGD and GRT of "Physics in Many-Sheeted Space-time".
23 comments:
So Keas idea is gone?
http://www.newscientist.com/article/dn21328-fasterthanlight-neutrinos-dealt-another-blow.html
"The only way to avoid this thing is to assume that, well, maybe on the way they went to other dimensions or something."
As usual, New Scientist does not make clear that the observation kills *only* the idea about tachyonicity as a reason why for neutrino velocity larger than light velocity.
It does not kill the explanations in which tachyonicity is not assumed: these however necessarily assume that light velocity is not maximal signal velocity or that maximal signal velocity depends on particle as in TGD framework.
The idea about tachyonic neutrinos fails in many ways but the failure described here is really fatal. If Kea believes on tachyonic neutrinos she is certainly forced to reconsider her beliefs.
Dear Matti,
I think there is a very analogies between algebra of m4*cp2 and geometric algebra! ;-)
Geometric algebra or Clifford geometric algebra regarded space time generated by four orthogonal vectors that make an algebra of G(1,3) with product of them with 16 term (4!=16). I think this algebra is something like hyperoctonion algebra (16 term), but geometric algebra makes it with a very geometric intuition. I very much enjoyed when I studied it for my BSC thesis. In geometric algebra an observer when measure some vector quantity or pseudovector (bivector in the approach), it doesn’t measure components of the quantity vector, but it measures components of a projective of components of the quantity on a subspace of G(1,3). is it like “Classical gauge potentials at space-time surface are obtained as projections of the components of CP2 spinor connection” ?
But in geometric algebra it is interpreted as:
Each observer sees a set of relative vectors that algebra of this relative vectors is a subalgebra of G(1,3). This is called a space-time split, and it is observer–dependent. This observer–dependent space-time and main space time relate together in a very beautiful manner. In first view one can feel then this is 8 dimensions (4+4) in general( something like your 8 dimensional space-time, but the approach doesn’t increase dimensions of space time more than 4. Because second space-time is observer–dependent and doesn’t should increase dimensions!
With Geometric algebra, much of the standard subject matter taught to physicists can be formulated in an elegant and highly condensed fashion. For example 4 Maxwell equations reduce to one in a beautiful manner.
For more details:
http://www.mrao.cam.ac.uk/~clifford/ptIIIcourse/GeometricAlgebraLectures.zip
Matti and all,
You may have noticed that in this discussion of superluminal neutrinos I used the special word teleomnic or tachyonic-like to describe such apparent phenomena.
The general theories, the supersymmetry of myself and Kea still stands, and always did.
The PeSla
Continue of my question:
In other word can we have a TGD theory without dimensions more than 4 with the approach of geometric algebra? ;-)
I know you have very strong arguments behind this Extra dimensions but in the approach of geometric algebra we have main 4 dimensions and other dimensions are in the form of bivectors(oriented plane segments) or trivectors(oriented volume segments) that is very intuitive.
http://www.newscientist.com/article/mg21328464.700-higgs-result-means-elegant-universe-is-back-in-vogue.html
Dear Hamed,
sorry for a slow response. To me it seems that geometric algebra is just 4-D (in complex sense) Clifford algebra, which accompanies Dirac equation.
Gamma matrices would be identifiable at given point of space-time as tangent vectors in the tangent space of space-time. What is the subspace of G(1,3)? This I did not understand this from your explanation.
The "subspace of relative vectors" seems to correspond to the quaternionic sub-space of octonionic space defining tangent space of space-time surface in TGD framework. Observer would see only associative sub-space. Space-time surface could be co-associative (normal space quaternionic and thus associative).
Partonic 2-surfaces and string would sheets would in turn correspond to commutative (co-commutative) sub-spaces.
In geometric algebra approach the use of quaternionic representation of gamma matrices would allow to identify string world sheets and partonic 2-surfaces in the similar manner as fundamental objects by the condition of commutative tangent space.
Clifford algebra in 8-D case has (complexified) octonionic variant for which gamma matrices are replaced with octonions and octonion product replaces matrix product. A possible manner to realize space-time dynamics purely number theoretically is as quaternionic 3-surface wold be to require that under this product induced gamma matrices form at every point quaternionic sub-algebra of octonionic algbra.
The big conjecture is that the preferred extremals of Kahler action can be identified as quaternionic/associative space-time surface in some sense- perhaps the above described sense.
Dear Hamed,
one could try to imagine a variant of TGD by taking 2-D surfaces in M^4 (or M^2 xS^2 as one sees) as basic objects: this would be the analog of string model. Something like follows.
*One could start from 2-surfaces in M^4 as the basic objects. In purely number theoretic approach the commutativity of local tangent space would define the dynamics.
Additional condition would be that each 2-D tangent space contains the preferred time direction corresponding to reals.
*With fixed time direction the direction of the selected imaginary unit (complex planes of quaternions is are parametrized by sphere) would characterize the local tangent space so that each 2-surface would define a field having values in sphere S^2. One might also speak about 2-surfaces M^2xS^2: M^4--M^2xS^2 duality could be a proper manner to say it.
*This would be like string model and one would obtain something resembling 2-D gravitation in this manner. S^2 carries U(1) spinor connection and one would obtain U(1) gauge field and one could ask whether there is a duality with a theory with 2-surfaces in M^2xS^2 being the basic objects.
*It seems very difficult to find non-trivial action principle for induced U(1) gauge field at 2-surfaces of M^2xS^2. Magnetic flux would be a purely topological action saying nothing about the interior of 2-D surface.
*At the1-D light-like curve serving as the counterpart of light-like 3-surface at which the signature of the induced metric changes from Minkowskian to Euclidian, the induced vector potential would be orthogonal to the light-like curve. In TGD framework same condition defines braid strands as Legendrean knots at light-like 3-surfaces. This would be a topological QFT.
In TGD framework one can proceed analogously.
*One can start from M^8 and require quaternionicity (associativity) or its co-property. One can pose the condition that each 4-D tangent space of space-time surface contains a preferred complex (commutative or co-commutative) plane. [This condition could be generalized to an integrable distribution of complex planes].
*This would give rise to space-time surface in M^8 and assignment of point of CP_2 to each point of it (S^2 point in above simplified case). This leads to a conjecture about M^8-M^4xCP_2 duality.
*One obtains 4-D gravitation and also a geometrization of electroweak gauge fields. Classical color gauge potentials are geometrized in terms of SU(2) Killing vector fields in CP_2. Projections again.
*The superposition of the effects of classical gauge fields (instead of the fields themselves) in the manner described also in this posting would allow to circumvent the basic objection.
*I mentioned already the Legendrean knots. Mayvbe TGD contains the simpler theory as a kind of sub-theory.
*This theory would be almost topological. Kahler action depending on metric would reduce by field equations and boundary conditions to Chern-Simons term but the dependence on metric would be there.
To Ulla:
I have heard many professionals to say 125 GeV Higgs mass means vacuum instability. I do not know under how general conditions this is true. There are indications for other scalars or pseudo-scalars: this was suddenly forgotten by bloggers when it became fashionable to believe that Higgs is found.
Standard SUSY is excluded to high and certainly one can still try to invent options allowing to save it. One might hope that after a year both standard SUSY and Higgsology are dead and buried and a progress in theory becomes finally possible.
One must just wait patiently: in absence of a real theory which could guide experimentalists, they must just slowly chop off heads of this thousan-headed dragon called phenomenological models. When one model is chopped out, two new equally silly models immediately appear.
Do Hamed:
I think that Clifford algebra shows its real power at the level of the "world of classical worlds" since it allows geometrization of fermionic oscillator operator algebra so that the dream of geometrizing entire quantum (field) theory becomes possible. This gives the connection with hyperfinite factors of type II_1 and with powerful mathematics of their inclusions and with quantum groups. I wish I could emphasize and underline enough the extreme importance of this geometrization.
.
Note that Kaehler structure allows geometrization of hermitian conjugation: without it one could not speak of fermionic creation and annihilation operators.
In super string models the analogs of WCW gamma matrices are obtained by making ordinary gamma matrices analogsof quantum fields. The fatal error is the assumption that they are hermitian requiring Majorana spinors: this leads to D=10 or 11 which was once thought to be breakthrough.
One fatal implication is the instability of proton and breakdown of separate conservation laws of B and L.
This variant of SUSY has now found to fail at LHC and it has been already proposed that N= 2 SUSY should replace it. N= 2 SUSY is indeed consistent with the 8-dimensional imbedding space (Majorana spinors are not possible).
Dear Matti,
So Thanks for clarifications. At some time I found an Approximation correspondence between multiplications of bases of geometric algebra of space-time (2^4 =16 term by two to two compounds and Triple compounds and others) and sedenions of Musean hypernumbers (16 term) and I think octonions and hyper-octonion and hyper-quaternions are subalgebra of the sedenions. I understood we can replace them instead of together and this lead to the same all of geometric algebra results ;-) .
Yet, something is unclear for me. In geometric algebra we regard all oriented plane segments (bivectors) and all oriented volume segments (trivectors) as vector spaces as like as oriented line segments and we can sum the bivectors together and trivectors together and even we can sum bivectors and trivectors and vectors together! These geometric concepts have strong algebraic meaning but lack of such concepts is obvious in your space time algebra (?) We doesn’t should these concepts as nonphysical concepts, these are like as oriented line segments (geometrical vectors) and if we consider bivectors, something the same as psudovectors(axial vectors) and aren’t new concepts, we are on incorrect way, because they are different in algebraic meaning and geometric meaning.
Can one consider algebraic meaning of extra dimensions of space-time as the same as algebraic meaning of bivectors or trivectors? (Considering them compose a vector space like as bases of extra dimensions). I think it doesn’t need considering the geometric meaning of extra dimensions (as extra dimensions of m4) to the algebraic meaning. In really, geometric meaning of bases of extra dimensions is the same as oriented plane segments or oriented volume segments.
If I am on an incorrect understanding, I’ll be grateful to you for guiding me.
Warning! The notions of hyperquaternion and hyper-octonion have several meanings.
a) I define them as sub-space of complexified quaterions/octonions. The additional imaginary unit is commutative. For sedenions it is not commutative! Sedensions are definitely not in question. One could of course consider also this possibility but I have not done this since I start from ordinary gamma matrix algebra with complex coefficients.
b) This structure is *not* an algebra but only a linear subspace.
c) The reason for this choice is that the norm squared defined by multiplying octonion and its conjugate (only octonionic imaginary units change sign) is Minkowskian. The products of hyper-complex gamma matrices are required to generate complexified quaternion algebra.
You are speaking about Clifford algebra. For its quaterninic/octononic variants the product of gammas is defined by using quaterionic/octonionic arihmetics and you get just gamma matrices. This is big difference. One must however notice that gammas in these representations are tensor products of Pauli sigma matrix sigma_2 and quaternion/octonion unit is that one can identify commutators of sigma matrices and they correspond to Lie algebra of G_2 rather than group SO(1,7).
Still one clarification.
*The octonionic variant of Clifford algebra is proposed only as an auxiality too to define what quaternionic space-time surface is and this is just a speculative possibility. I have not been able to prove or disprove it but intuitive feeling is tht this is "must be true".
*An alternative definition would be based on real-octonion analyticity. The surfaces obtained by putting the "imaginary" part of this kind of function to zero is 4-D. The proposal or rather- the question is- whether this gives quaternionic surfaces and that these could provide the preferred extremals of Kahler action.
This would be an immensely beautiful structure. One can add and multiply these functions and composites fog are possible. These operations would be therefore defined also for space-time surfaces!
Finally answer to your question: In TGD framework the algebraic meaning of extra dimensions is not the same as that of bi- and trivectors. Clifford algebra has 8 generators.
One might think instantons with a self dual Maxwell field at infinity, were not physically relevant. However, one can promote them to being Einstein Yang Mills solutions, with a constant self dual Yang Mills field at infinity. One could then match them to Yang Mills instantons in flat space, with large winding numbers, which can have regions where the Yang Mills field, is almost constant.
Infinite-D hierarchy of spacetime sheets? Instanton is ike a very big plasmoid.
http://www.hawking.org.uk/gravitational-entropy.html
http://www.hawking.org.uk/rotation-nut-charge-and-anti-de-sitter-space.html
The quarter area law, holds for black holes or black branes in any dimension, d, that have a horizon, which is a d minus 2 dimensional fixed point set, of a U1 isometry group. However Chris Hunter and I, have recently shown that entropy can be associated with a more general class of space-times. In these, the U1 isometry group can have fixed points on surfaces of any even co-dimension, and the space-time need not be asymptotically flat, or asymptotically anti de Sitter. In these more general class, the entropy is not just a quarter the area, of the d minus two fixed point set.
Among the more general class of space-times for which entropy can be defined, an interesting case is those with Nut charge. Nut charge can be defined in four dimensions, and can be regarded as a magnetic type of mass.
Is this really the famous scientist? No famous crackpot? Magnetic mass? Oh my! And black branes!
About the LHC plasma phase? Link to a neutron star? http://www.astronomynow.com/news/n1010/28neutronstar/
http://news.discovery.com/space/dark-matter-web-mapped-120109.html#mkcpgn=fbnws1
remember the possible dark matter galaxies?
Manysheeted spacetime,
http://arxiv.org/abs/1009.1136
Carlip says recent work in loop quantum gravity, high temperature string theory, renormalization group analysis applied to general relativity and other areas of quantum gravity research, all hints at a two dimensional spacetime on the smallest scale. In most of these cases, the number of dimensions simply collapse in a process called spontaneous dimensional reduction as the scale reduces.
One obvious question is that if only two dimensions are present on this scale, which two are they? Carlip calculates that they must be one of time and one of space. "At each point, the dynamics picks out a "preferred" spatial direction, leading to approximately (1+1)-dimensional local physics," he says.
To Ulla:
In TGD effective 2-dimensionality is prediction of strong form of general coordinate invariance and realizes holography in strong sense.
What Carlip suggests has nothing to do with this. I must say that I share the attitude of Lubos towards quantum loop gravity. The fatal failure of loop quantum gravity is that it neglects all other interactions and starts from the purely mathematical ideas related to canonical quantization of general relativity, which is mathematical non-sense.
Super string model approach does not make this failure but is led to wrong track by the belief that the basic objects are 2-D in 10 or 11 D space. The correct question would have been "How to generalized conformal invariance to higher dimensions" and the answer would have been "replace strings with light-like 3-surfaces". Second correct question would have been "How one could fix the imbedding space (to avoid landscape problem) so that one obtains standard model symmetries: the answer would have been M^4xCP_2". In Matrix version of M-theory the basic objects are points and situation gets hopeless. Sad.
Einstein started from essentially philosophical vision. The general problem of superstring and loop gravity approaches and needless to say - most other approaches - is the total lack of "philosophy".
There are obvious candidates for deeper guiding principles. For instance, the generalization of Einstein's program to quantum physics as infinite-D geometry suggests itself strongly but both stringers and loopers have tried to build the a theory of everything on basis of ad hoc guesses and failed miserably.
Nima Arkani Hamed seems to realize the need for a big vision and guiding principles. He has a very nice talk (see Peter Woit's page for a link) in which he used 40 minutes to explain "philosophy".
Emergent space-time is an attempt to the direction of deeper principle and also Nima suggests this. Emergence has however generated a lot of non-sense since no-one knows what "emergent" could mean: only Muenchausen could define this notion. Nima however correctly identifies the basic problem: the lack of local observables in quantum gravity.
And now comes the need for a precise thinking! This does not require giving up space-time! It only suggests that lower dimensional objects carry the information about quantum states. Partonic 2-surfaces at light-like boundaries of CD defining zero energy states in TGD framework.
Also one should ask what one means with quantum state. Ordinary ontology assumes quite too much. In zero energy ontology gives rise to a generalization of S-matrix to M-matrix and U-matrix (Nima has also realized that unitary S-matrix is not enough) and also a generalization of Yangian invariance and new hugely simplified view about Feynman diagrams.
Holography is also excellent deep idea but its AdS/CFT realization leads to the wrong track. In TGD strong form of holography follows from strong form of general coordinate invariance as also the correct identification of observables: Einstein already showed us the way but we are too stupid to follow his advise!
http://www.philica.com/display_article.php?article_id=42
a lightray x.s. see the Z_0 and massless extremals?
Holography
http://www.nobelprize.org/nobel_prizes/physics/laureates/1971/gabor-lecture.pdf
It's interesting to notice that the German Wikipedia articles title many of the founding fathers of QT etc. big names both Physiker and Philosopher, where as Enlgish articles call same persons only physicists. Steven Weinberg's "Dreams of a Final Theory" has a whole chapter titled "Against Philosophy" (after defending his metaphysics of ontological reductionism ;)). It's a cultural thing and the fitting motto of the anti-philosophical "pragmatism" which has also it's totalitarian and authoritarian aspects: "Shut up and calculate! :D
A question: how do you see the relation between Alain Connes' approach and TGD?
Dear Anonymous,
to my view this "anti-philosophical" attitude explains 40 wasted years in theoretical physics. One can climb to the tree only via root and in physics philosophy is the root.
There are several connections to the Connes's work. I am of course not a mathematian so that I am not able to say anything about technicalities. I try to list what comes into my mind.
1. Alan Connes is creator of non-commutative geometry. He is also a specialist in von Neuman algebras: hyper-finite factors (HFFs) of type III_1 appear in his work and relate to quantum field theories. He is probably the correct person if you want (and dare, I probably would not;-)) ask something about inclusions of HFFs.
Possible connection:
a) The spinors of "world of classical worlds" (very roughly fermionic Fock states for partonic 2-surface or collection of them) are HFFS of type II_1 and there are could reasons to believe that HFFs of type III_1 are also unavoidable in TGD framework as one considers spinor fields of WCW instead of spinors.
b) Finite measurement resolution is described naturally in terms of inclusions: included algebra acts like gauge algebra generating states not distinguishable in finite measurement resolution. The resulting coset structure would be finite fractional-dimensional non-commutative geometry. Quantum spinors would be a good example.
2. Connes has suggested that HFFs appear in mathematical formulation of statistical physics and there is unique one parameter flow associated with them which he wants to interpret physically.
Possible connection:
a) In TGD U matrix the fundamental object. It is unitary and its rows are "complex square roots" of density matrices rho: hermitian square root of density matrix times unitary S-matrix. These objects are analogous to square roots of exponent of Hamiltonian in thermodynamics.
The Hermitian square roots form an infinite-D Lie algebra and M-matrices in turn generate Kac-Moody type algebra in powers of S if these hermitian matrices commute with S-matrix. The objects representing zero energy states act as their own symmetry algebra which should generalize the Yangian algebra about which Nima Arkani Hamed talks enthusiastically!
b) The needs of TGD would require taking the "square root" of the existing mathematical theory of the basic objects associated with HFFs of type III_1.
3. Quantum groups relate to non-commutative geometry and also to HFFs and therefore also to TGD. In TGD framework the interpretation is in terms of finite measurement resolution. I do not know about Connes's own interpretation. Usually quantum groups are assigned with Planck length scale.
4. In Connes's unification proposal the generalization of spinor and Dirac operator is essential. I understand that he wants to replace Riemannian view about the notion of length with quantal view in which infinitesimal length ds somehow relates to Dirac operator- maybe ds corresponds to finite-D operator of Hilbert space being infinitesimal for hyper-finite factors for which all finite-D projection operators have finite trace and unit operator has unit trace. I admit that I failed to understand this idea at deeper level.
Possible connection: In TGD framework quantum states are modes of classical WCW spinor fields so that fermionic statistics is geometrized.
In the well known interview (http://www.ipm.ac.ir/IPM/news/connes-interview.pdf) Connes sounds like nice guy who likes heretics. You have many times said that TGD would need help from a mathematician and you share similar interests (space-time and primes etc) and attitudes, so what do you have to lose if you ask him a couple question? I don't think being shy is your real problem, but being afraid of hierarchies, and that's only healthy.
I am certainly afraid of hierarchies and for full reason! But I also feel myself desperately inferior in relation to mathematicians like Connes. I know my strengths gut they are different from those of mathematicians like Connes and those of first rate theoretical physicists. I am simply not super-technicians as they are.
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