Friday, December 12, 2014
Sunday, November 9, 2014
PARTICLES IN QUANTUM GRAVITY
The purpose of this post is to explain a recent discovery that we did with my two physicists collaborators Ali Chamseddine and Slava Mukhanov. We wrote a long paper Geometry and the Quantum: Basics which we put on the arXiv, but somehow I feel the urge to explain the result in non-technical terms.
The subject is the notion of particle in Quantum Gravity. In particle physics there is a well accepted notion of particle which is the same as that of irreducible representation of the Poincaré group. It is thus natural to expect that the notion of particle in Quantum Gravity will involve irreducible representations in Hilbert space, and the question is "of what?".
What we have found is a candidate answer which is a degree 4 analogue of the Heisenberg canonical commutation relation [p,q]=ih. The degree 4 is related to the dimension of space-time. The role of the operator p is now played by the Dirac operator D. The role of q is played by the Feynman slash of real fields, so that one applies the same recipe to spatial variables as one does to momentum variables. The equation is then of the form E(Z[D,Z]^4)=\gamma where \gamma is the chirality and where the E of an operator is its projection on the commutant of the gamma matrices used to define the Feynman slash.
Our main results then are that:
1) Every spin 4-manifold M (smooth compact connected)
appears as an irreducible representation of our two-sided equation.
2) The algebra generated by the slashed fields is the algebra of functions on M
with values in A=M_2(H)\oplus M_4(C), which is exactly the slightly noncommutative
algebra needed to produce gravity coupled to the Standard Model minimally
extended to an asymptotically free theory.
3) The only constraint on the Riemannian metric of the 4-manifold is that its volume
is quantized, which means that it is an integer (larger than 4) in Planck units.
The result 1) is a consequence of deep results in immersion theory going back to the work of Smale, and also to geometric results on the construction of 4-manifolds as ramified covers of the 4-sphere, where the optimal result is a result of Iori and Piergallini asserting that one can always assume that the ramification occurs over smooth surfaces and with 5 layers in the ramified cover. The dimension 4 appears as the critical dimension because finding a given manifold as an irreducible representation requires finding two maps to the sphere such that their singular sets do not intersect. In dimension n the singular sets can have (as a virtue of complex analysis) dimension as low as n-2 (but no less) and thus a general position argument works if (n-2)+(n-2) is less than n, while n=4 is the critical value.
The result 2) is a consequence of the classification of Clifford algebras. When working in dimension 4, the sphere lives in five dimensional Euclidean space and to write its equation as the sum of squares of the five coordinates one needs 5 gamma matrices. The two Clifford algebras Cliff(+,+,+,+,+) and Cliff(-,-,-,-,-) are respectively M_2(H)+ M_2(H) and M_4(C). Thus taking an irreducible representation of each of them yields respectively M_2(H) and M_4(C).
The result 3) comes from the index formula in noncommutative geometry. One shows that the degree 4 equation implies that the volume of the manifold (which is defined as the leading term of the Weyl asymptotics of the eigenvalues of the Dirac operator) is the sum of two Fredholm indices and is thus an integer. It relies heavily on the cyclic cohomology index formula and the determination of the Hochschild class of the Chern character.
The great advantage of 3) is that, since the volume is quantized, the huge cosmological term which dominates the spectral action is now quantized and no longer interferes with the equations of motion which as a result of our many years collaboration with Ali Chamseddine gives back the Einstein equations coupled with the Standard Model.
The big plus of 2) is that we finally understand the meaning of the strange choice of algebras that seems to be privileged by nature: it is the simplest way of replacing a number of coordinates by a single operator. Moreover as the result of our collaboration with Walter van Suijlekom, we found that the slight extension of the SM to a Pati-Salam model given by the algebra M_2(H)\oplus M_4(C) greatly improves things from the mathematical standpoint while moreover making the model asymptotically free! (see Beyond the spectral standard model, emergence of Pati-Salam unification.)
To get a mental picture of the meaning of 1), I will try an image which came gradually while we were
working on the problem of realizing all spin 4-manifolds with arbitrarily large quantized volume as a
solution to the equation.
"The Euclidean space-time history unfolds to macroscopic dimension from the product of two 4-spheres of Planckian volume as a butterfly unfolds from its chrysalis."
The subject is the notion of particle in Quantum Gravity. In particle physics there is a well accepted notion of particle which is the same as that of irreducible representation of the Poincaré group. It is thus natural to expect that the notion of particle in Quantum Gravity will involve irreducible representations in Hilbert space, and the question is "of what?".
What we have found is a candidate answer which is a degree 4 analogue of the Heisenberg canonical commutation relation [p,q]=ih. The degree 4 is related to the dimension of space-time. The role of the operator p is now played by the Dirac operator D. The role of q is played by the Feynman slash of real fields, so that one applies the same recipe to spatial variables as one does to momentum variables. The equation is then of the form E(Z[D,Z]^4)=\gamma where \gamma is the chirality and where the E of an operator is its projection on the commutant of the gamma matrices used to define the Feynman slash.
Our main results then are that:
1) Every spin 4-manifold M (smooth compact connected)
appears as an irreducible representation of our two-sided equation.
2) The algebra generated by the slashed fields is the algebra of functions on M
with values in A=M_2(H)\oplus M_4(C), which is exactly the slightly noncommutative
algebra needed to produce gravity coupled to the Standard Model minimally
extended to an asymptotically free theory.
3) The only constraint on the Riemannian metric of the 4-manifold is that its volume
is quantized, which means that it is an integer (larger than 4) in Planck units.
The result 1) is a consequence of deep results in immersion theory going back to the work of Smale, and also to geometric results on the construction of 4-manifolds as ramified covers of the 4-sphere, where the optimal result is a result of Iori and Piergallini asserting that one can always assume that the ramification occurs over smooth surfaces and with 5 layers in the ramified cover. The dimension 4 appears as the critical dimension because finding a given manifold as an irreducible representation requires finding two maps to the sphere such that their singular sets do not intersect. In dimension n the singular sets can have (as a virtue of complex analysis) dimension as low as n-2 (but no less) and thus a general position argument works if (n-2)+(n-2) is less than n, while n=4 is the critical value.
The result 2) is a consequence of the classification of Clifford algebras. When working in dimension 4, the sphere lives in five dimensional Euclidean space and to write its equation as the sum of squares of the five coordinates one needs 5 gamma matrices. The two Clifford algebras Cliff(+,+,+,+,+) and Cliff(-,-,-,-,-) are respectively M_2(H)+ M_2(H) and M_4(C). Thus taking an irreducible representation of each of them yields respectively M_2(H) and M_4(C).
The result 3) comes from the index formula in noncommutative geometry. One shows that the degree 4 equation implies that the volume of the manifold (which is defined as the leading term of the Weyl asymptotics of the eigenvalues of the Dirac operator) is the sum of two Fredholm indices and is thus an integer. It relies heavily on the cyclic cohomology index formula and the determination of the Hochschild class of the Chern character.
The great advantage of 3) is that, since the volume is quantized, the huge cosmological term which dominates the spectral action is now quantized and no longer interferes with the equations of motion which as a result of our many years collaboration with Ali Chamseddine gives back the Einstein equations coupled with the Standard Model.
The big plus of 2) is that we finally understand the meaning of the strange choice of algebras that seems to be privileged by nature: it is the simplest way of replacing a number of coordinates by a single operator. Moreover as the result of our collaboration with Walter van Suijlekom, we found that the slight extension of the SM to a Pati-Salam model given by the algebra M_2(H)\oplus M_4(C) greatly improves things from the mathematical standpoint while moreover making the model asymptotically free! (see Beyond the spectral standard model, emergence of Pati-Salam unification.)
To get a mental picture of the meaning of 1), I will try an image which came gradually while we were
working on the problem of realizing all spin 4-manifolds with arbitrarily large quantized volume as a
solution to the equation.
"The Euclidean space-time history unfolds to macroscopic dimension from the product of two 4-spheres of Planckian volume as a butterfly unfolds from its chrysalis."
Tuesday, August 26, 2014
Wednesday, August 13, 2014
Fields Medals 2014: Maryam Mirzakhani, Martin Hairer, Manjul Bhargava, Artur Avila
Congratulations to all 2014 Fields medalists! Very well deserved and also really nice to see a woman wining a Fields medal for the first time ever (and of course I am specially delighted that she has the same undergraduate alma mater, Sharif University, as I! Quanta magazine has a coverage of all four winners Avila, Bhargava, Hairer, Mirzakhani.
It was a bit unusual to see the results announced by IMU before the opening ceremonies! You can follow all the discussions and news from here.
It was a bit unusual to see the results announced by IMU before the opening ceremonies! You can follow all the discussions and news from here.
Saturday, July 5, 2014
Lectures on Video
I would like to draw your attention to the following lectures just posted on youtube
1. Alain Connes: Arithmetic Site
Update: and a related interview where some of the relevant ideas in topos theory and the impact of Grothendieck is discussed.
2. Ali Chamseddine: Spectral Geometric Unification
1. Alain Connes: Arithmetic Site
Update: and a related interview where some of the relevant ideas in topos theory and the impact of Grothendieck is discussed.
2. Ali Chamseddine: Spectral Geometric Unification
Thursday, June 5, 2014
Announcement book "Noncommutative Geometry and Particle Physics" by Walter van Suijlekom
My book "Noncommutative Geometry and Particle Physics" is due to appear this summer with Springer:
This textbook provides an introduction to noncommutative geometry and presents a
number of its recent applications to particle physics. It is intended
for graduate students in mathematics/theoretical physics who are new to
the field of noncommutative geometry, as well as for researchers in
mathematics/theoretical physics with an interest in the physical
applications of noncommutative geometry. In the first part, we introduce
the main concepts and techniques by studying finite noncommutative
spaces, providing a “light” approach to noncommutative geometry. We then
proceed with the general framework by defining and analyzing
noncommutative spin manifolds and deriving some main results on them,
such as the local index formula. In the second part, we show how
noncommutative spin manifolds naturally give rise to gauge theories,
applying this principle to specific examples. We subsequently
geometrically derive abelian and non-abelian Yang-Mills gauge theories,
and eventually the full Standard Model of particle physics, and conclude
by explaining how noncommutative geometry might indicate how to proceed
beyond the Standard Model.
Wednesday, June 4, 2014
Tuesday, March 18, 2014
Review of a paper by Gebhard Boeckle and the group S_(q)
So this post is a bit of an experiment. My friends at Math Reviews recently sent me a really interesting Math. Z. paper by Gebhard Boeckle. I spent some time reviewing it and found it contained very interesting results and calculations that pointed, yet again, to some possible underlying action of the group $S_{(q)}$ that I have discussed in other posts here. If you combine it with the new results of Rudy Perkins in http://arxiv.org/abs/1402.4000, the situation becomes even more intriguing....
The review at Math Reviews has number MR3127039. Here is the link: http://www.ams.org/mathscinet-getitem?mr=3127039
With the concurrence of MR, and with my sincere gratitude, I am posting the review below; if you find it too small, you can simply increase the size of the font (by something like 'command +'). (I should also say that I converted the original pdf to jpegs which could then be uploaded to Blogger...)
The review at Math Reviews has number MR3127039. Here is the link: http://www.ams.org/mathscinet-getitem?mr=3127039
With the concurrence of MR, and with my sincere gratitude, I am posting the review below; if you find it too small, you can simply increase the size of the font (by something like 'command +'). (I should also say that I converted the original pdf to jpegs which could then be uploaded to Blogger...)
Tuesday, February 4, 2014
zeta zeroes AND gamma poles
The arithmetic of function fields over finite fields has always been a ``looking-glass'' window into the standard arithmetic of number fields, varieties, motives etc.; sort of ``life based on silicon'' as opposed to the classical ``carbon-based'' complex-valued constructions. It has constantly amazed me, and frankly given me great pleasure, to see the way that analogies always seem to work out in one form or another. Often these analogies are not at all obvious and I want to report here on the existence of a certain analogy that I find particularly satisfying and greatly encouraging.
One of my great desires in working in this area was, and of course is, to have a fully analytic theory of $L$-series in characteristic $p$ based on Drinfeld modules, $t$-modules, etc., (as opposed to the fundamentally algebraic nature of the complex valued functions traditionally defined for function fields). For a long time we have known the correct definitions of Euler factors at the good primes and, with the work of Gardeyn, we also know the correct definitions at the bad places (at least in the case of Drinfeld modules). We further know that these $L$-series have excellent analyticity properties with associated ``trivial zeroes''; moreover, in the simplest case of $A:=F_q[\theta]$, the infinite prime, and the associated zeta function, we know that the zeroes are actually are simple and ``lie on the line'' $F_q((1/\theta))$. Until now these trivial zeroes arose by using the (polynomial) Euler factors at infinity coming from classical theory, or cohomology of crystals, etc., and some auxiliary arguments.
More recently, beginning with the work of Taelman and Lafforgue, there has been really exciting progress in establishing the ``correct'' analogs of the class group and class number formulae in this context. Indeed, this an area of great current excitement and active research. See for instance: http://hal.archives-ouvertes.fr/hal-00940567 .
Given these very strong indications, it is not unreasonable to expect that many, if not all, of the remaining properties from the complex (``carbon-based'') $L$-series should ultimately show up in some form or other in the finite characteristic theory. So, in this post I will briefly describe an observation about trivial zeroes due to Rudy Perkins, and based on the wonderful preprint http://arxiv.org/abs/1301.3608v2 of Bruno Angles and Federico Pellarin, which shows, yet again, the remarkable similarities between the classical theory of $L$-series and their finite characteristic cousins.
As every arithmetician knows, in order to truly appreciate the analytic properties of classical $L$-series (of number fields) one must adjoin to them a finite number of Euler factors at the infinite primes. These Euler factors are, of course, created out of Euler's fabulous gamma function $\Gamma (s)$. And everybody knows that $\Gamma (s)$ is nowhere zero with simple poles at the nonpositive integers. Via the functional equation of a given $L$-series, these poles translate into the fundamental ``trivial zeroes'' of the $L$-series (which often times may also be deduced in a more elementary fashion) as well as determining the exact order of these zeroes.
In the finite characteristic case, using the Carlitz exponential and factorial, I was able to define a number of continuous (and even rigid analytic) $\Gamma$-analogs which capture many of the properties of Euler's $\Gamma(s)$ (due to the fundamental work of Greg Anderson, Dinesh Thakur, Dale Brownawell, Matt Papanikolas,...). However, there was no obvious connection with $L$-series or their trivial zeroes (which were originally obtained using the polynomial Euler factors associated to the infinite primes as mentioned above). Of course, as can be imagined, this was truly a disappointment.
On the other hand, beginning with their fundamental work on tensor powers of the Carlitz module, Greg Anderson and Dinesh Thakur introduced another $\Gamma$-analog denoted $\omega (t)$. This is a nowhere zero function with simple poles at {$\theta$qj} for $j\geq 0$. Subsequently, this function proved to be instrumental in studying the properties of the previously mentioned gamma functions. When I first saw it, I noticed how natural it seemed as a deformation of the Carlitz period (a $\Gamma$-type property after all!). However, I also found the collection of poles of $\omega(t)$ too specialized to somehow be related to $L$-series; in this I was simply wrong (for which I am grateful!).
The reason I was wrong is due to the fundamental work of Federico Pellarin over the past few years. Federico introduced the natural (but seemingly highly non-classical) set {$\chi$t} of quasi-characters of $A$ given simply by the maps $f(\theta)\mapsto f(t)$ where $t$ is some constant. He then naturally defines the $L$-series $L(\chi$t,s) and, most importantly, establishes a wonderful formula relating the special values of these $L$-series to the Carlitz period and $\omega(t)$. Federico also made the elementary but totally key observation that $L(\chi$u, s)=$\zeta$(s-qj), where $u=\theta$qj.
Thus, the poles of $\omega(t)$ are completely canonical, and actually represent the qj-th power morphisms on $A$. I can't help but wonder if there is a some sort of similar interpretation of the poles of Euler's $\Gamma(s)$.
Still, what about $\zeta(s-i)$ for any positive $i$?????
Obviously [C : R]=2; but R and C are the only local fields with the property that their algebraic closure comprises a finite dimensional extension. For function fields over finite fields, the
algebraic closure of the associated local fields are vast objects with a huge amount of ``room to move.'' Put more directly, one can simply add quasi-characters at will and consider $L$-series of the form $L(\chi$t1,...,$\chi$te, s) for arbitrary e. This gives a staggering and bewildering (at least with the current state of the art) amount of flexibility, but it really does work and one can indeed clearly specialize (in many ways) to $\zeta(s-i)$. In other words, it is mandatory to adjoin an arbitrary number of $\Gamma$-factors to a fixed $L$-series.
And this brings us back to the preprint http://arxiv.org/abs/1301.3608v2 of Bruno and Federico. Here a beautiful integrality result, Theorem 4, is obtained for $L(\chi$t1 ...$\chi$te, $\alpha$) and
$\Pi \omega($ti) where $\alpha$ is a positive integer and $\alpha\equiv e$ mod (q-1).
Finally to close the circle of arguments, Rudy Perkins has just shown me a quick and elegant argument, by specializing the {ti}, how this Theorem 4 implies the existence of trivial zeroes in great generality (and certainly those of $\zeta(s)$ at the negative integers divisible by $q-1$). Briefly here is what Rudy does: Given an integer s divisible by q-1, one uses s+1 and 1 in this theorem; on the left one then has an expression involving the $L$-function (viewed as a function of the {ti}) and on the right one has a multi-variable polynomial (which is the "integrality" part of the result). Upon taking the limit of the last variable at $\theta$, one obtains that the value in question times $\Pi$ $\omega$(ti) is still a polynomial. But the value in question can easily be seen to also be a polynomial and it must have zeros all over the place in order to cancel the $Gamma$-poles. So many zeroes, in fact, that it identically vanishes!
The order of these trivial zeroes is another matter. While one can compute these orders using elementary arguments in certain cases, a more deeper approach now seems truly to be indicated...
(Added2-6-2014: Lenny Taelman has produced some highly valuable notes of his Beijing lectures and these are now in a form, while still preliminary, that can be shared: please see
http://www.math.leidenuniv.nl/~lenny/beijing.pdf
Also along these very same lines, please see the preprint by Jiangxue Fang
http://arxiv.org/abs/1401.1293v1 )
(Added 2-18-2014: Perkins' paper "An exact degree for multivariate special polynomials" is on the arXiv at http://arxiv.org/abs/1402.4000 .)
One of my great desires in working in this area was, and of course is, to have a fully analytic theory of $L$-series in characteristic $p$ based on Drinfeld modules, $t$-modules, etc., (as opposed to the fundamentally algebraic nature of the complex valued functions traditionally defined for function fields). For a long time we have known the correct definitions of Euler factors at the good primes and, with the work of Gardeyn, we also know the correct definitions at the bad places (at least in the case of Drinfeld modules). We further know that these $L$-series have excellent analyticity properties with associated ``trivial zeroes''; moreover, in the simplest case of $A:=F_q[\theta]$, the infinite prime, and the associated zeta function, we know that the zeroes are actually are simple and ``lie on the line'' $F_q((1/\theta))$. Until now these trivial zeroes arose by using the (polynomial) Euler factors at infinity coming from classical theory, or cohomology of crystals, etc., and some auxiliary arguments.
More recently, beginning with the work of Taelman and Lafforgue, there has been really exciting progress in establishing the ``correct'' analogs of the class group and class number formulae in this context. Indeed, this an area of great current excitement and active research. See for instance: http://hal.archives-ouvertes.fr/hal-00940567 .
Given these very strong indications, it is not unreasonable to expect that many, if not all, of the remaining properties from the complex (``carbon-based'') $L$-series should ultimately show up in some form or other in the finite characteristic theory. So, in this post I will briefly describe an observation about trivial zeroes due to Rudy Perkins, and based on the wonderful preprint http://arxiv.org/abs/1301.3608v2 of Bruno Angles and Federico Pellarin, which shows, yet again, the remarkable similarities between the classical theory of $L$-series and their finite characteristic cousins.
As every arithmetician knows, in order to truly appreciate the analytic properties of classical $L$-series (of number fields) one must adjoin to them a finite number of Euler factors at the infinite primes. These Euler factors are, of course, created out of Euler's fabulous gamma function $\Gamma (s)$. And everybody knows that $\Gamma (s)$ is nowhere zero with simple poles at the nonpositive integers. Via the functional equation of a given $L$-series, these poles translate into the fundamental ``trivial zeroes'' of the $L$-series (which often times may also be deduced in a more elementary fashion) as well as determining the exact order of these zeroes.
In the finite characteristic case, using the Carlitz exponential and factorial, I was able to define a number of continuous (and even rigid analytic) $\Gamma$-analogs which capture many of the properties of Euler's $\Gamma(s)$ (due to the fundamental work of Greg Anderson, Dinesh Thakur, Dale Brownawell, Matt Papanikolas,...). However, there was no obvious connection with $L$-series or their trivial zeroes (which were originally obtained using the polynomial Euler factors associated to the infinite primes as mentioned above). Of course, as can be imagined, this was truly a disappointment.
On the other hand, beginning with their fundamental work on tensor powers of the Carlitz module, Greg Anderson and Dinesh Thakur introduced another $\Gamma$-analog denoted $\omega (t)$. This is a nowhere zero function with simple poles at {$\theta$qj} for $j\geq 0$. Subsequently, this function proved to be instrumental in studying the properties of the previously mentioned gamma functions. When I first saw it, I noticed how natural it seemed as a deformation of the Carlitz period (a $\Gamma$-type property after all!). However, I also found the collection of poles of $\omega(t)$ too specialized to somehow be related to $L$-series; in this I was simply wrong (for which I am grateful!).
The reason I was wrong is due to the fundamental work of Federico Pellarin over the past few years. Federico introduced the natural (but seemingly highly non-classical) set {$\chi$t} of quasi-characters of $A$ given simply by the maps $f(\theta)\mapsto f(t)$ where $t$ is some constant. He then naturally defines the $L$-series $L(\chi$t,s) and, most importantly, establishes a wonderful formula relating the special values of these $L$-series to the Carlitz period and $\omega(t)$. Federico also made the elementary but totally key observation that $L(\chi$u, s)=$\zeta$(s-qj), where $u=\theta$qj.
Thus, the poles of $\omega(t)$ are completely canonical, and actually represent the qj-th power morphisms on $A$. I can't help but wonder if there is a some sort of similar interpretation of the poles of Euler's $\Gamma(s)$.
Still, what about $\zeta(s-i)$ for any positive $i$?????
Obviously [C : R]=2; but R and C are the only local fields with the property that their algebraic closure comprises a finite dimensional extension. For function fields over finite fields, the
algebraic closure of the associated local fields are vast objects with a huge amount of ``room to move.'' Put more directly, one can simply add quasi-characters at will and consider $L$-series of the form $L(\chi$t1,...,$\chi$te, s) for arbitrary e. This gives a staggering and bewildering (at least with the current state of the art) amount of flexibility, but it really does work and one can indeed clearly specialize (in many ways) to $\zeta(s-i)$. In other words, it is mandatory to adjoin an arbitrary number of $\Gamma$-factors to a fixed $L$-series.
And this brings us back to the preprint http://arxiv.org/abs/1301.3608v2 of Bruno and Federico. Here a beautiful integrality result, Theorem 4, is obtained for $L(\chi$t1 ...$\chi$te, $\alpha$) and
$\Pi \omega($ti) where $\alpha$ is a positive integer and $\alpha\equiv e$ mod (q-1).
Finally to close the circle of arguments, Rudy Perkins has just shown me a quick and elegant argument, by specializing the {ti}, how this Theorem 4 implies the existence of trivial zeroes in great generality (and certainly those of $\zeta(s)$ at the negative integers divisible by $q-1$). Briefly here is what Rudy does: Given an integer s divisible by q-1, one uses s+1 and 1 in this theorem; on the left one then has an expression involving the $L$-function (viewed as a function of the {ti}) and on the right one has a multi-variable polynomial (which is the "integrality" part of the result). Upon taking the limit of the last variable at $\theta$, one obtains that the value in question times $\Pi$ $\omega$(ti) is still a polynomial. But the value in question can easily be seen to also be a polynomial and it must have zeros all over the place in order to cancel the $Gamma$-poles. So many zeroes, in fact, that it identically vanishes!
The order of these trivial zeroes is another matter. While one can compute these orders using elementary arguments in certain cases, a more deeper approach now seems truly to be indicated...
(Added2-6-2014: Lenny Taelman has produced some highly valuable notes of his Beijing lectures and these are now in a form, while still preliminary, that can be shared: please see
http://www.math.leidenuniv.nl/~lenny/beijing.pdf
Also along these very same lines, please see the preprint by Jiangxue Fang
http://arxiv.org/abs/1401.1293v1 )
(Added 2-18-2014: Perkins' paper "An exact degree for multivariate special polynomials" is on the arXiv at http://arxiv.org/abs/1402.4000 .)
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