Noncommutative Geometry for Quantum Spacetime

Transcription

1 Noncommutative Geometry for Quantum Spacetime The view from below Fedele Lizzi Università di Napoli Federico II Institut de Ciencies del Cosmo, Barcelona SIGrav, Alessandria 2014

2 I think we would all agree that there is the need for new physics at very high energy, say at least at the order of Planck s scale, if not before. The aim of this seminar is to argue that we will also need a new geometry Why do we need it? The old one, based on points, lines, vector fields, curvature tensors and the like, has served us quite well for a few millennia We already have a Noncommutative Geometry And we call it quantum mechanics 1

3 I think we would all agree that there is the need for new physics at very high energy, say at least at the order of Planck s scale, if not before. The aim of this seminar is to argue that we will also need a new geometry Why do we need it? The old one, based on points, lines, vector fields, curvature tensors and the like, has served us quite well for a few millennia We already have a Noncommutative Geometry And we call it quantum mechanics 2

4 I think we would all agree that there is the need for new physics at very high energy, say at least at the order of Planck s scale, if not before. The aim of this seminar is to argue that we will also need a new geometry Why do we need it? The old one, based on points, lines, vector fields, curvature tensors and the like, has served us quite well for a few millennia We already have a Noncommutative Geometry And we call it quantum mechanics 3

5 A naive way to see that in quantum mechanics the concept of point (of phase space) is not valid is given by the Heisenberg Microscope The idea is that to see something small, of size of the order of x, we have to send a small photon, that is a photon with a small wavelength λ, but a small wavelength means a large momentum p = h/λ. In the collision there will a transfer of momentum, so that we can capture the photon. The amount of momentum transferred is uncertain. In quantum mechanics a point in phase space is an untenable concept because of the Heisenberg uncertainty principle: x p 2 4

6 We know how to handle the problem. Position and momenta, and all observables, become operator on a Hilbert space (which represent the particle content of the theory), and operators may be noncommutative. But we had to abandon the possibility of localizing the particle at a point in phase space. So far we have been discussing the noncommutativity of phase space. In quantum mechanics however configuration space is still an ordinary space. So it is in general relativity Is it legitimate to expect the usual geometry to hold to all scales? As far as I know the first to consider the possibility of noncommutative spacetime was Heisenberg who, in a letter to Peierls in 1930, expressed the hope that a noncommutative structure might eliminate the infinities of quantum field theory, which at that time were considered a problem. 5

7 In 1935 Bronstein observed that the combination of general relativity and quantum mechanics might create problems for the localization of events. At the same caricatural level used earlier: In order to measure the position of an object, and hence the point in space, one has use a very small probe, and quantum mechanics forces us to have it very energetic, but on the other side general relativity tells us that if too much energy is concentrated in a region a black hole is formed. 6

8 This is the region in which the theory we would like to use is Quantum Gravity. Unfortunately a theory we do not yet have. In fact the two problems are related. A quantum gravity theory needs spacetime to be a different object from the one used in classical geometry. A solution of the localization problem is given by spacetime with noncommuting coordinates, whose commutator is a central operator. This was introduced by Doplicher-Fredenhagen-Roberts. Note however that several theories of quantum gravity have a spacetime which at some level has a noncommutative structure. this is true for Loop Quantum Gravity, but also in string theory a deformed noncommutative product of fields appeared (Seiberg-Witten) 7

9 As a first attempt we can replicate the quantum mechanical form of noncommutatity [x µ, x ν ] = iθ µν Describe the product of fields via a deformed (Gronewöld-Moyal) product originally introduced for quantum mechanics. f g = fe2 i θµν µ ν g Noncommutativity of spacetime is in the deformation of the algebra of fields. Something like: L = µ φ µ φ m2 2 φ φ λ 4! φ 4 8

10 It is possible to apply this deformation of space also to gravity theories. But consider Symmetries The -product (and its variants), because of the presence of θ µν, breaks Lorentz invariance. On the other side a Moyal product with covariant derivatives is not associative But if we quantize spacetime, we should also quantize the symmetries: Quantum groups and Hopf Algebras The presence of a deformed symmetry gives a way to deform in a consistent way all the geometry, a procedure pioneered by the late Julius Wess and collaborators. Our hosts here (Aschieri and Castellani) have developed this formalism and found the presence of extended theories of gravitation in the form oh higher order (in θ ) corrections to the Einstein Hilbert space 9

11 The view form below But I want to dare more, and use the knowledge I have form field theory to say something about properties of spacetime given the present high energy data. In particular it is clear that quantization of spacetime is a deep ultraviolet problem In the following I will use the tools given by the spectral approach to noncommutative geometry mostly pioneered by Connes. I will use the knowledge form field theory at energies below the (yet to be defined) transition scale at which quantum geometry appears, to infer some knowledge of quantum spacetime We are having new data from high energy experiments, mainly LHC, so this is a good moment to explore the consequences of field theory. The way one can learn what happens beyond the scale of an experiment is to use the renormalization flow of the theory 10

12 We know that the coupling constants, i.e. change with energy. the strength of the interaction, 11

13 This picture is valid in the absence of new physics, i.e. new particles and new interactions which would alter the equations which govern the running The three interaction strength start from rather different values but come together almost at a single unification point But then the nonabelian interactionsprocced towards asymptotic freedom, while the abelian one climbs towards a Landau pole at incredibly high energies GeV The lack of a unification point was one of the reasons for the falling out of fashion of GUT s. Some supersymmetric theories have unification point. 12

14 We all believe that this running will be stopped by something at GeV This unknown something we call quantum gravity I take the point of view that there is a fundamental change of the degrees of freedom of spacetime at, or before the Planck scale, and that the tools to describe this are the one of Noncomutative Spectral Geometry Following this paradigm, for me the topological nature of spacetime is in the spectrum of the algebra of functions on spacetime (or its generalization), and everything is described by operators acting on some Hilbert space, which is natural to consider composed by the fermionic matter fields 13

15 We need now to add the information about the metric properties of spacetime, and this is done via the Dirac operator D, or its suitable generalizations. The Dirac operator knows a lot about the space on which it is defined, its square is the Laplacian, but it also has information about the spin structure, and acting on gauge fields it carries implicit information also about the gauge vector bosons. Most of modern geometry is based on the properties of the Dirac operator, and in particular the eigenvalues of the Dirac operator on a curved spacetime are diffeomorphism-invariant functions of the geometry. They form an infinite set of observables for general relativity. For me therefore D = γ m u( µ + ω µ ), with ω the spin connection 14

16 There is a way to describe interaction of fields in a background using the Chamseddine-Connes Spectral Action S = Tr χ ( D 2 A Λ 2 χ is a cutoff function, which we may take to be a decreasing exponential or the characteristic function of the interval: χ(x) = { 1 x 1 0 x > 1 ) D A = D + A is a fluctuation of the Dirac operator, A a connection oneform built from D as A = i a i[d, b i ] with a, b elements of the algebra, the fluctuations are ultimately the variables, the fields of the action Λ is a cutoff scale. I will consider it to have the physical meaning of a transition scale, to a yet unknown phase. 15

17 The spectral action can be expanded in powers of Λ 2 using heat kernel. In this framework it is possible to describe the standard model in a gravitational background. As algebra of fields one takes functions valued in an appropriate finite dimensional algebra representing the gauge group. One has to choose as D operator the tensor product of the usual Dirac operator on a curved background / times a matrix containing the fermionic parameters of the standard model (Yukawa couplings and mixings), acting on the Hilbert space of fermions In this way one saves one parameter, and can predict the mass of the Higgs. The original prediction was 170 GeV, which is not a bad result considering that the theory is basically based on pure mathematical requirements When it was found at 125 GeV refined models with right handed neutrinos make it compatible with present experiments. (Stephan, Devastato Martinetti, FL, Chamseddine, Connes, Van Suijlekom But this is another seminar... 16

18 Technically the bosonic spectral action is a sum of residues and can be expanded in a power series in terms of Λ 1 as S B = n f n a n (D 2 A /Λ2 ) where the f n are the momenta of χ f 0 = f 2 = 0 0 dx xχ(x) dx χ(x) f 2n+4 = ( 1) n n x χ(x) x=0 n 0 the a n are the Seeley-de Witt coefficients which vanish for n odd. For D 2 A of the form D 2 = (g µν µ ν I + α µ µ + β) 17

19 Defining (in term of a generalized spin connection containing also the gauge fields) then a 0 = Λ4 16π 2 a 2 = Λ2 16π 2 ω µ = 1 2 g ( µν α ν + g σρ Γ ν σρ I) Ω µν = µ ω ν ν ω µ + [ω µ, ω ν ] E = β g ( µν µ ω ν + ω µ ω ν Γ ρ µν ω ) ρ dx 4 g tr I F dx 4 g tr ( R6 ) + E dx 4 g tr ( 12 µ µ R + 5R 2 2R µν R µν R µνσρ R µνσρ 60RE + 180E µ µ E + 30Ω µν Ω µν ) a 4 = 1 16π 2 1 tr is the trace over the inner indices of the finite algebra A F and Ω and E contain the gauge degrees of freedom including the gauge stress energy tensors and the Higgs, which is given by the inner fluctuations of D 18

20 Let me dwell on the role of Λ. The spectral action uses it as a regulator. Without it the trace diverges. This points to a geometry in which the Dirac operator is actually a truncated operator, i.e. which saturates at Λ the Dirac operator has a spectrum of increasing eigenvalues Consider the eigenvectors n of D respective eigenvalue λ n. in increasing order of the Call N the number of eigenvalues smaller then Λ, i.e. λ n < Λ for n < N, zero otherwise. 19

21 Truncation of the spectrum can be made by the projection operator P Λ = n n n N Defining D Λ = DP Λ + (1 P Λ ) Λ, P Λ = Θ(Λ 2 D 2 ) We are effectively saturating the operator at a scale Λ 20

22 Given a space with a Dirac operator one can define a distance (Connes) between states of the algebra of functions, in particular points are (pure) states and the distance is: d(x, y) = sup f(x) f(y) [D,f] 1 It is possible possible to prove D Andrea, FL, Martinetti that using D Λ the distance among points is infinite. But the distance among smeared points, like coherent spates, may be finite. In general for a bounded Dirac operator of norm Λ then d(, x, y) > Λ 1, and to find states at finite distance one has to consider extended points, such as coherent spates 21

23 The role of a truncated Dirac operator, or in general wave operator, has been introduced in field theory even before the sacral action. It goes under the name of Finite Mode Regularization, Andrianov, Bonora, Fujikawa Consider the generic fermionic action: Z = [d ψ][dψ]e ψ D ψ formally = det D The equality is formal because the expression is divergent, and has to be regularized, for example considering D Λ 22

24 One can study the renormalization flow, and note that the measure is not invariant under scale transformation, giving rise to a potential anomaly Andrianov, Kurkov, FL The induced term by the flow, which takes care of the anomaly, turn out to be exactly the spectral action The hypothesis is that Λ has a physical meaning, it is a scale indicating a phase transition, and we can try to infer some properties of the phase above Λ studying the high energy limit of the action with the cutoff. At high momentum Green s function, the inverse of D Λ, effectively is the identity in momentum space I will now see this in greater detail considering the bosonic sector 23

25 Usually probes are bosons, hence let me consider the expansion of the spectral action in the high momentum limit Kurkov, FL, Vassilevich This has been made by Barvinsky and Vilkovisky who were able to sum all derivatives (for a decreasing exponential cutoff function): Tr exp ( D2 Λ 2 Λ 4 ( R µν f 1 ( P f 3 ( 2 Λ 2 ) 2 Λ 2 Λ4 ) R + P f 4 ( (4π) d 4 2 x g tr [ 1 + Λ 2 P + ) ( ) R µν + Rf 2 2 R+ 2 Λ 2 Λ 2 ) P + Ω µν f 5 ( 2 Λ 2 ) Ω µν)] + O(R 3, Ω 3, E 3 ) where P = E R and f 1,..., f 5 are known functions, high momenta asymptotic of form factor: 24

26 f 1...f 5 read: f 1 (ξ) 1 6 ξ 1 ξ 2 + O ( ξ 3) f 2 (ξ) 1 18 ξ ξ 2 + O ( ξ 3 ) f 3 (ξ) 1 3 ξ ξ 2 + O ( ξ 3) f 4 (ξ) ξ ξ 2 + O ( ξ 3) f 5 (ξ) 1 2 ξ 1 ξ 2 + O ( ξ 3) 25

27 Let me consider a Dirac operator containing just the relevant aspects, i.e. a bosonic fields and the fluctuations of the metric. /D = iγ µ µ + γ 5 φ = iγ µ ( µ + ω µ + ia µ ) + γ 5 φ with ω µ the Levi-Civita connection and g µν = δ µν + h µν It is now possible to perform the B-V expansion to get the expression for the high energy spectral action S B Λ4 (4π) 2 d 4 x [ 3 2 h µνh µν + 8φ 1 ] 2φ + 8F 1 µν ( 2 ) 2F µν 26

28 In order to understand the meaning of this action let me remind how we get propagation of waves and correlation of points in usual QFT with action S[J, ϕ] = d 4 x [ ϕ(x) ( 2 + m 2) ϕ(x) J(x)ϕ(x) ] To this correspond the equation of motion ( 2 + m 2) ϕ(y) = J(y) And the Green s function G(x y) which propagates the source: ϕ J (x) = d 4 yj(y)g(x y) 27

29 In momentum representation we have ϕ(x) = 1 (2π) 2 d 4 k e ikx ˆϕ(k) J(x) = 1 (2π) 2 d 4 k e ikx Ĵ(k) G(x y) = 1 (2π) 2 d 4 k e ik(x y) Ĝ(k) And the propagator is G(k) = 1 ( k 2 + m 2) The field at a point depends on the value of field in nearby points, and the points talk to each other exchanging virtual particles 28

30 In the general case of a generic boson ϕ, the Higgs, an intermediate vector boson or the graviton ( ) S[J, φ] = d 4 1 x 2 ϕ(x)f ( 2 )ϕ(x) J(x)ϕ(x), In this case the equation of motion is F ( 2 )φ(x) = J(x) giving G = 1 F ( 2 ), G(k) = 1 F ( k 2 ) and ϕ J (x) = d 4 yj(y)g(x y) = 1 (2π) 4 d 4 ke ikx 1 J(k) F ( k 2 ) 29

31 The cutoff is telling us that ϕ J (x) = d 4 yj(y)g(x y) = 1 (2π) 4 d 4 ke ikx 1 J(k) F ( k 2 ) The short distance behaviour is given by the limit k Consider J(k) 0 for k 2 [K 2, K 2 + δk 2 ], with K 2 very large. ϕ J (x) K 1 (2π) d k 4 e ikx J(k)k 2 = ( 2 )J(x) 1 (2π) d k 4 e ikx J(k) = J(x) for scalars and vectors for gravitons 30

32 This corresponds to a limit of the Green s function in position space G(x y) { ( 2 )δ (x y) for scalars and vectors δ (x y) for gravitons The correlation vanishes for noncoinciding points, heuristically, nearby points do not talk to each other. This is a limiting behaviour, I think one has to take it as a general indication that the presence of a physical cutoff scale in momenta leads to a non geometric phase in which the concept of point ceases to have meaning, possibly described by a noncommutative geometry Note that throughout this discussion I have done nothing to spacetime, I have only imposed the cutoff and used standard techniques and interpretations 31

33 Conclusions It is very difficult to explain to a deep water fish the concept of air Still he will know that as he goes up the pressure will decrease And he can also infer some properties of a different states of matter by looking at bubbles which are creates near some high energy volcanic vents or when above there are storms What I presented here is hopefully just an element of a larger picture which should use gravity in a fundamental way I am convinced that what has to change at high scales is the very geometry of spacetime, and this work is but an indication that the direction is the loss of correlation among the points 32

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