the de sitter universe - univie.ac.at€¦ · klein-gordon tachyons iii (schroer) • feinberg’s...
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de Sitter Tachyons(at the LHC Era)
Rigorous QFT at the LHC era. ESI-Vienna. September 28, 2011
Ugo MoschellaUniversità dell’Insubria, Como, Italia – SPhT Saclay
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Tachyons
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Relativistic Tachyons
• Is the trasmission of energy faster than light contradicting special relativity?
• When , E and p diverge.
• A classical massive particle moving at one time cannot move at a later time; this does not rule out other “particles” moving faster. Photons for instance do exist! But they are massless
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Relativistic Tachyons II
A
AB
B
A
Feinberg’s reinterpretation principle: Emission of a positive energy tachyon = Absorption of a negative energy tachyon
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The degrees of freedom of standard linear fields are provided by
the commutator function: a bidistribution that vanishes at spacelike
separated pairs
On a globally hyperbolic manifold the equations of motion plus canonical initial conditions uniquely determine the commutator
Linear Quantum Field Theory
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A Hilbert space representation is associated to any two-point
function that solves the equations of motion
The functional equation (canonical quantization)
And the positive-definiteness condition
For Minkowski,dS, adS or other simmetries if unbroken: invariance:
Two-point functions
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There exist a complete set of states having positive energy
1) 2 pt functions and propagators
3) Perturbation theory 4) Renormalization
2) Free fields
equivalent to
Spectral Property
Consequences
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Klein-Gordon field: a crash course
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Klein-Gordon bradyons
Spectral condition
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The spectral condition implies that the Fourier representation is meaningful in a domain
of the complex Minkowski spacetime
)V z(Imz T
)V z(Imz T
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The Cut reflects Causalityand the Quantum
Spectrum + Lorentz Inv.
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Klein-Gordon tachyons I
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Klein-Gordon tachyons I
• The 2-p function is covariant and positive definite
• The commutator vanishes identically
• The anticommutator does not (but the quantization is not canonical)
• The energy spectrum is unbounded below
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Klein-Gordon tachyons II (Feinberg)
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Klein-Gordon tachyons II (Feinberg)
• The “quantization” is not covariant but positive• The commutator does not vanishes; it is neither canonical nor
covariant• The anticommutator is covariant but it not canonical (Feinberg
invokes the name “scalar fermions” for this. )• The energy spectrum is bounded below in every frame. Positive in
the chosen frame (non-covariant spectral condition)
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Klein-Gordon tachyons III (Schroer)
• Feinberg’s quantization may be rendered covariant by adding a complex manifold to Feinberg’s integration cycle
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Klein-Gordon tachyons III (Schroer)
• Feinberg’s quantization may be rendered covariant by adding a complex manifold to Feinberg’s integration cycle
• The 2-point function is covariant but not positive definite. It is not a tempered distribution but grows exponentially in momentum space
• Locality holds. The theory is canonical• There exists no Lorentz invariant physical positive subspace• This quantization may be obtained as the analytic continuation to
imaginary masses of the standard KG fields
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Summary
• When quantizing the Klein-Gordon tachyon it is impossible to reconcile the axioms.
• Give up either locality and covariance or positivity (and of course the spectral condition).
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de Sitter
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Supernovae Era SN1997 ff
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Supernova SN1997 ff
SN 1997 → Sidereus Nuncius 1997
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Energy content of the universe
dark energy 74%
Deark Matter21%
Ordinary matter 5%
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Eq. Friedmann
74% 21%
5%
Today
99% 1%
Tomorrow
+ State Eqs. od the various components of the cosmic fluid
= time evolution of the Hubble “constant”
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Einstein Equations
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Spherical de Sitter model
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Flat de Sitter model
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Open de Sitter model
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The shape of our universe
M(d+1) : ημν = diag(1,-1,……,-1)
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The asymptotic cone
M(d+1) : ημν = diag(1,-1,……,-1)
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The asymptotic cone: causal structure
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ds timelike geodesics
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The asymptotic cone as the de Sitter momentum space
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Conserved quantities
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Energy
• In special relativity the energy of a particle is measured relative to an arbitrary given Lorentz frame, being the zero component of a four-vector.
• This picture does not extend to the de Sitter case where frames are defined only locally.
• The maximal symmetry of the de Sitter universe allows for the energy of a pointlike particle to be defined relative to just one reference massive free particle understood conventionally to be at rest (a sharply localized observer)
• The energy of the free particle (ξ,η) with respect to the reference geodesic (u,v) is defined as follows
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Energy
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Energy
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Energy
• E can be interpreted as the correct de Sitter energy of the particle is confirmed by noting that it is the conserved quantity associated to the invariance of the particle action under time translation.
• Indeed, since in flat coordinates the spatial distances dilate in the course of time by the exponential factor the expression of an infinitesimal symmetry under time evolution is
• The action
• is invariant and E can be obtained from S by standard methods
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Classical scattering
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Example: particle decay
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de Sitter plane waves
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de Sitter plane waves
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Principal de Sitter waves
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Complementary de Sitter waves
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Discrete de Sitter waves
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dS: construction of two-point functions
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They are dS invariant
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The plane waves are however irregular
skip
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Spectral condition: de Sitter tubes
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are globally well-defined in the tubes.
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Fourier representation for BD 2-point functions
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Fourier representation on the real manifold
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= -1
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Physical fields
• For principal fields positivity is easy
• de Sitter Fourier transform(s)
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Complementary fields (inflation)
• Positivity is a little less direct. For ν>0 we have
• The kernel in the following expression is positive definite for 0< ν<(d-1)/2
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De Sitter Tachyons
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The field equation gets an anomaly
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Euclidean approach (Folacci)
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Zero modes
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Infrared divergence
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BRS quantization
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BRS quantization
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Fourier representation
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Sketch of the proof
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Remarks
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Conclusion and Outlook
• Physics of the de Sitter Tachyons remains to be understood
• This work was done in collaboration and friendship with Jacques Bros and Henri Epstein
• See you in Vienna again!