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Bell violation in the Sky
Department of Theoretical Physics
Date: 13/09/2016
Contact: [email protected]
Webpage: http://tifr.academia.edu/sayantanchoudhury
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Thanks to……………………
All present and past Collaborators………..
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Thanks to……………………Rajeev Singh, Pune University.
Pratik Chatterjee, IISER-Mohali.
Sandeep Dash, Siddhant K. Panda, NISER, Bhubaneswar.
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In Collaboration with………………
Prof. SudhakarPanda
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Special thanks to ………………
Prof. Sandip P. TrivediProf. Gautam MandalProf. Guruprasad Kar
Based on
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Key references • J. Maldacena, A model with cosmological
Bell inequalities, Fortsch.Phys. 64 (2016) 10-23, arXiv:1508.01082 [hep-th].
• N. Arkani Hamed, J. Maldacena, Cosmological Collider Physics,arXiv:1503.08043[hep-th].
• S. Panda, Y. Sumitomo, S. P. Trivedi, Axionas Quintessence in String Theory, Phys.Rev. D83 (2011) 083506, arXiv:1011.5877[hep-th].
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Key references • A. Einstein,B. Podolsky, N. Rosen, Can
quantum mechanical description of physical reality be considered complete, Phys.Rev. 47 (1935) 777-780.
• J. S. Bell, On the Einstein-Podolsky-Rosen paradox, Physics 1 (1964) 195-200.
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Other useful references • P. Svrcek, E. Witten, Axions in String Theory,
JHEP 0606 (2006) 051, arXiv:hep-th/0605206.
• E. Silverstein, A. Westphal, Monodromy in the CMB: Gravity Waves and String Inflation,Phys.Rev. D78 (2008) 106003, arXiv:0803.3085[hep-th].
• R. Flauger, L. McAllister, E. Pajer, A. Westphal, G. Xu, Oscillations in the CMB from AxionMonodromy Inflation, JCAP 1006 (2010) 009, arXiv:0907.2916[hep-th].
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• Introduction.
• Bell’s inequality in QM.
• Cosmological Bell violating setup.
• Role of massive (new) particles.
• Analogy with axion fluctuations from String Theory.
• Role of isospin breaking interactions.
• Role of spin (s=0,s=2,s>2).
• Role of decoherence.
• Bottom lines and future prospects.10
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Introduction
• It is well known that according to the inflationary paradigm, primordial fluctuations seen in CMB were produced by quantum mechanical effects in the early universe.
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• It is well known that according to the inflationary paradigm, primordial fluctuations seen in CMB were produced by quantum mechanical effects in the early universe.
• Such fluctuations provide seed for large scale structure formation.
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• It is well known that according to the inflationary paradigm, primordial fluctuations seen in CMB were produced by quantum mechanical effects in the early universe.
• Such fluctuations provide seed for large scale structure formation.
• The fluctuations that we observe at present are classical in nature.
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• During the quantum mechanical interpretation of the required fluctuations, highly entangled quantum mechanical wave function of the universe plays a significant role.
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• During the quantum mechanical interpretation of the required fluctuations, highly entangled quantum mechanical wave function of the universe plays a significant role.
• Due to this fact, quantum fluctuations can be theoretically demonstrated as well as implemented in the context of primordial cosmology, iff we can perform a Bell's inequality violating cosmological
experiment using the highly quantum mechanical
entangled wave function of the universe defined in
the inflationary period. 16
• To describe the background methodology it is important to mention that, in the context of
quantum mechanics, Bell test experiment is
described by the measurement of two non-
commutating physical operators which are
associated with two distinctive locations in the
space-time.
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• To describe the background methodology it is important to mention that, in the context of
quantum mechanics, Bell test experiment is
described by the measurement of two non-
commutating physical operators which are
associated with two distinctive locations in the
space-time.
• Similarly using this similar analogy in the context of primordial cosmology, one can also perform similar cosmological observations on two spatially separated as well as causally disconnected places upto the epoch of reheating.
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• In case of cosmological observations one can able to measure the numerical values of various cosmological observables (along with cosmic variance), which can be computed from scalar curvature fluctuation.
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• In case of cosmological observations one can able to measure the numerical values of various cosmological observables (along with cosmic variance), which can be computed from scalar curvature fluctuation.
• Apart from the success in its observational ground it is important to point that for all such observations it is impossible to measure the value of associated canonically conjugate momentum.
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• In case of cosmological observations one can able to measure the numerical values of various cosmological observables (along with cosmic variance), which can be computed from scalar curvature fluctuation.
• Apart from the success in its observational ground it is important to point that for all such observations it is impossible to measure the value of associated canonically conjugate momentum.
• Consequently, for these observables it is impossible to measure the imprints of two non-commuting operators in primordial cosmology.
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• But, if the cosmological observables are satisfying the basic requirements of decoherence effect then it is possible to perform measurements from two exactly commuting cosmological observables and one can able to design a Bell's inequality violating cosmological experimental setup.
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• But, if the cosmological observables are satisfying the basic requirements of decoherence effect then it is possible to perform measurements from two exactly commuting cosmological observables and one can able to design a Bell's inequality violating cosmological experimental setup.
• In quantum mechanics, to design such experimental setup one needs to perform repeated measurement on the same object (here it is the same quantum state) and in such a physical situation one can justify the appearance of each and every measurement through a single quantum state.
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• In the context of cosmology, one can similarly consider two spatially separated portions in the full sky which exactly mimics the role of performing repeated cosmological Bell's inequality violating experiment via the same quantum mechanical state.
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• In the context of cosmology, one can similarly consider two spatially separated portions in the full sky which exactly mimics the role of performing repeated cosmological Bell's inequality violating experiment via the same quantum mechanical state.
• Due to this here one can choose the appropriate and required properties of two spatially separated portions in the full sky to setup Bell's inequality violating experimental setup in cosmology.
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• In the context of cosmology, one can similarly consider two spatially separated portions in the full sky which exactly mimics the role of performing repeated cosmological Bell's inequality violating experiment via the same quantum mechanical state.
• Due to this here one can choose the appropriate and required properties of two spatially separated portions in the full sky to setup Bell's inequality violating experimental setup in cosmology.
• If it is possible to connect direct a link between these mentioned non-commuting cosmological observables and a classical probability distribution function originated from inflationary paradigm then it is surely possible to setup a Bell's inequality violating cosmological experimental setup.
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Bell’s inequality in QM
• Concept of reality = Real properties of microscopic objects determines the results of quantum mechanical experiments,
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• Concept of reality = Real properties of microscopic objects determines the results of quantum mechanical experiments,
• Concept of locality = That reality in one place is not affected by experiments done at the same time at a distant place.
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• Concept of reality = Real properties of microscopic objects determines the results of quantum mechanical experiments,
• Concept of locality = That reality in one place is not affected by experiments done at the same time at a distant place.
Bell’s inequality=No local hidden variable theory is compatible with quantum mechanics.- John Stewart Bell (1964).
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• If we choose:
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• If we choose:
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• If we choose:
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θ=𝜋/4For we get the maximal violation of the Bell’s inequality, which contains an
extra 2 factor.
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Cosmological Bell violating
setup
• Each Fourier mode of fluctuation can be treated as a time dependent harmonic oscillator.
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• Each Fourier mode of fluctuation can be treated as a time dependent harmonic oscillator.
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• Each Fourier mode of fluctuation can be treated as a time dependent harmonic oscillator.
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• Each Fourier mode of fluctuation can be treated as a time dependent harmonic oscillator.
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• Each Fourier mode of fluctuation can be treated as a time dependent harmonic oscillator.
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This implies that fluctuation becomes classicalwhen they exit the horizon.
• After inflation when reheating occurs one can write down a measure or more precisely a probability distribution function of fluctuation:
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• After inflation when reheating occurs one can write down a measure or more precisely a probability distribution function of fluctuation:
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Q. Can we really able to distinguishbetween this probability distribution
from a purely classical distribution function?
• After inflation when reheating occurs one can write down a measure or more precisely a probability distribution function of fluctuation:
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Q. Can we really able to distinguishbetween this probability distribution
from a purely classical distribution function?
Testing Quantum Mechanics
• After inflation when reheating occurs one can write down a measure or more precisely a probability distribution function of fluctuation:
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Q. Can we really able to distinguishbetween this probability distribution
from a purely classical distribution function?
1.Many many successful predictions in Quantum Mechanics.
• After inflation when reheating occurs one can write down a measure or more precisely a probability distribution function of fluctuation:
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Q. Can we really able to distinguishbetween this probability distribution
from a purely classical distribution function?
1.Many many successful predictions in Quantum Mechanics.
2. Fundamental deviation from the classical theory (probability distribution) via Bell’s inequality violation.
Strategy
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Strategy• Choose a toy model of universe that will make the
job easy.
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Strategy• Choose a toy model of universe that will make the
job easy.
• Here we are not at claiming that this is the unique model of universe using which one can design the setup.
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Strategy• Choose a toy model of universe that will make the
job easy.
• Here we are not at claiming that this is the unique model of universe using which one can design the setup.
• Since we don’t have any direct observational evidence we also can’t claim that this toy model is our known universe. But may be in future this will be tested.
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Strategy• Choose a toy model of universe that will make the
job easy.
• Here we are not at claiming that this is the unique model of universe using which one can design the setup.
• Since we don’t have any direct observational evidence we also can’t claim that this toy model is our known universe. But may be in future this will be tested.
• In this toy model of universe we test the validity of Bell’s inequality with primordial fluctuations.
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Strategy• In this computation massive particles with spin “s”
(s=0,2 and >2 allowed) and additionally “isospin” quantum number plays important role.
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Strategy• In this computation massive particles with spin “s”
(s=0,2 and >2 allowed) and additionally “isospin” quantum number plays important role.
• We provide the example for Stringy axion (s=0 scalar) which has “isospin”.
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Strategy• In this computation massive particles with spin “s”
(s=0,2 and >2 allowed) and additionally “isospin” quantum number plays important role.
• We provide the example for Stringy axion (s=0 scalar) which has “isospin”.
• Time dependent mass profile with dependence in “isospin” makes the job easy here. Axions have such profile.
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Strategy• In this computation massive particles with spin “s”
(s=0,2 and >2 allowed) and additionally “isospin” quantum number plays important role.
• We provide the example for Stringy axion (s=0 scalar) which has “isospin”.
• Time dependent mass profile with dependence in “isospin” makes the job easy here. Axions have such profile.
• Such time dependence in mass profile of the massive particles (ex. axion) produces classical perturbations on the inflaton. As a result hot spots produced in CMB by curvature fluctuations and all such massive particles are visible today. 59
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Role of massive (new)
particles
• Massive particle pair creation is required to produce hot and cold spots in CMB.
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• Massive particle pair creation is required to produce hot and cold spots in CMB.
• Massive particle mass should have time dependent profile. Growing mass profile is usually preferred. But some times other behavior is also important depending on the specific mathematical structure of the mass profile.
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• Massive particle pair creation is required to produce hot and cold spots in CMB.
• Massive particle mass should have time dependent profile. Growing mass profile is usually preferred. But some times other behavior is also important depending on the specific mathematical structure of the mass profile.
• Heavy fields become heavy at early and late times, but in an intermediate time scale where inflation occurs it behaves like a light field.
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• Massive particle pair creation is required to produce hot and cold spots in CMB.
• Massive particle mass should have time dependent profile. Growing mass profile is usually preferred. But some times other behavior is also important depending on the specific mathematical structure of the mass profile.
• Heavy fields become heavy at early and late times, but in an intermediate time scale where inflation occurs it behaves like a light field.
• Each massive particle pair is created in “isospin” singlet manner. This is required for measurement in the detectors.
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• Eqn of motion for massive field:
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• Eqn of motion for massive field:
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• Eqn of motion for massive field:
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Exact solution only possible if weassume that thetime dependencein mass profileis very slow.
We choose Bunch Davies and α vacuum.
• Eqn of motion for massive field:
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Exact solution only possible if weassume that thetime dependencein mass profileis very slow.
We choose Bunch Davies and α vacuum.
WKB approximated solutionfor arbitrary time dependent mass profile
Bogoluibov co-efficient
• Eqn of motion for massive field:
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Exact solution only possible if weassume that thetime dependencein mass profileis very slow.
We choose Bunch Davies and α vacuum.
WKB approximated solutionfor arbitrary time dependent mass profile
Bogoluibov co-efficient
• Toy model 1:
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• Toy model 1:
• Toy model 2:
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• Toy model 1:
• Toy model 2:
• Toy model 3:
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Distribution of massive particles from Toy model 1
• Curvature fluctuations in presence of massive particles:
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• Curvature fluctuations in presence of massive particles:
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One point function
• Curvature fluctuations in presence of massive particles:
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One point function
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One point function
One point function
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One point function
One point function
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Two pointfunction
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Two pointfunction
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Two pointfunction
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Two pointfunction
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Two pointfunction
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Two pointfunction
Important Note• Scale of inflation and the associated new physics
can be predicted without the prior knowledge of primordial gravity waves in a model independent way if the amount of Bell violation is known.
• Here we only need to measure the mass of the particles participating in evolution of universe and for serve this purpose we need to measure the one point function scalar fluctuations.
• Additionally, it is important to mention here that the scale of inflation can expressed in terms of known inflationary observables for proper choice of initial condition (vacua).
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Important Note• In case of single field slow-roll models of inflation
the amount of non-Gaussianity (three point function) is proportional to the first slow-roll parameter or more precisely with the primordial gravity waves through tensor-toscalar ratio “r”. Using the methodology one can derive the model independent expression for the non-Gaussian amplitude in terms of the time dependent mass.
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Analogy with axion
fluctuations from String Theory
• Axion monodromy model:
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• Axion monodromy model:
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• Axion monodromy model:
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• Axion monodromy model:
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Scale of the effective potential:
• Axion monodromy model:
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Scale of the effective potential:
Warp factor:
• Axion monodromy model:
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Scale of the effective potential:
Warp factor:
String Scale:
• Axion have time dependent decay constant. It serves the purpose of time dependent mass for heavy particles as mentioned earlier.
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• Axion have time dependent decay constant. It serves the purpose of time dependent mass for heavy particles as mentioned earlier.
• Time dependent decay constant is large at early and late times. It becomes small during some time, a few e-foldings after creation of massive axion pairs. Then it becomes large.
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• Axion have time dependent decay constant. It serves the purpose of time dependent mass for heavy particles as mentioned earlier.
• Time dependent decay constant is large at early and late times. It becomes small during some time, a few e-foldings after creation of massive axion pairs. Then it becomes large.
• Fluctuations are produced at the characteristic scale of axions.
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• Axion fluctuation with time dependent decay constant:
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• Axion fluctuation with time dependent decay constant:
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Toy model for Axion decay constant profile
• Axion fluctuation with time dependent decay constant:
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Toy model for Axion decay constant profile
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Role of isospin breaking
interaction
• Isospin breaking interaction:
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• Isospin breaking interaction:
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• Isospin breaking interaction:
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• Isospin breaking interaction:
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• Isospin breaking interaction:
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Mass eigen value
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Toy model for mass parameter
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Toy model for mass parameter
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Toy model for mass parameter
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Toy model for mass parameter
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Toy model for mass parameter
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Toy model for mass parameterNote:*No breaking of isospin at early times.**Distinctive values of the mass eigenvalues implies hot distinguishable spots in CMB.
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Distinguishable hot spots
Pair of massive particleswith +1 and -1 eigenvalues.
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Role of spin(s=0,s=2,
s>2)
• Eqn of motion for heavy field with spin S:
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• Eqn of motion for heavy field with spin S:
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WKB approximated solution
• Bound on mass parameter:
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• Bound on mass parameter:
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For more ….
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Future prospects and Bottom lines
Bottom lines
• We provide a toy model for Bell’s inequality violation in cosmology.
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Bottom lines
• We provide a toy model for Bell’s inequality violation in cosmology.
• Model consist of inflaton and additional massive field with time dependent behaviour.
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Bottom lines
• We provide a toy model for Bell’s inequality violation in cosmology.
• Model consist of inflaton and additional massive field with time dependent behaviour.
• For each model massive particle creation in “isospin” singlet state plays crucial role.
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Bottom lines
• We provide a toy model for Bell’s inequality violation in cosmology.
• Model consist of inflaton and additional massive field with time dependent behaviour.
• For each model massive particle creation in “isospin” singlet state plays crucial role.
• Prescribed methodology is consistent with axionfluctuations appearing in the context of String Theory.
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Bottom lines• We provide a toy model for Bell’s inequality
violation in cosmology.
• Model consist of inflaton and additional massive field with time dependent behaviour.
• For each model massive particle creation in “isospin” singlet state plays crucial role.
• Prescribed methodology is consistent with axionfluctuations appearing in the context of String Theory.
• Signature of the Bell violation is visualized from non-zero one point function of curvature fluctuation.
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Future Prospect
• Effect on three point function (primordial non-Gaussianity) for curvature fluctuations in presence of Bell’s inequality violation.
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Future Prospect
• Effect on three point function (primordial non-Gaussianity) for curvature fluctuations in presence of Bell’s inequality violation.
• Explicit role of quantum entanglement and decoherence.
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Future Prospect
• Effect on three point function (primordial non-Gaussianity) for curvature fluctuations in presence of Bell’s inequality violation.
• Explicit role of quantum entanglement and decoherence.
• Connection with quantum information.
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Future Prospect
• Effect on three point function (primordial non-Gaussianity) for curvature fluctuations in presence of Bell’s inequality violation.
• Explicit role of quantum entanglement and decoherence.
• Connection with quantum information.
• Explicit role in presence of stochastic fluctuations and random noise. 132
Future Prospect
• Effect on three point function (primordial non-Gaussianity) for curvature fluctuations in presence of Bell’s inequality violation.
• Explicit role of quantum entanglement and decoherence.
• Connection with quantum information.
• Explicit role in presence of stochastic fluctuations and random noise. 133
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Thanks for your time……
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Back up slides
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Role of decoherence
• Basis field eigenstates satisfies:
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• Basis field eigenstates satisfies:
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Quantum operatorspecified in anypoint in thespace-time
Eigenvalue represents the classical configuration
• Basis field eigenstates satisfies:
• Arbitrary vacuum state:
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Quantum operatorspecified in anypoint in thespace-time
Eigenvalue represents the classical configuration
• Basis field eigenstates satisfies:
• Arbitrary vacuum state:
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Quantum operatorspecified in anypoint in thespace-time
Eigenvalue represents the classical configuration
Coherentsuperposition Of Eigen states
• Basis field eigenstates satisfies:
• Arbitrary vacuum state:
• To accommodate decoherence we consider a general environment G(x) (non-linier) in the interaction:
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Quantum operatorspecified in anypoint in thespace-time
Eigenvalue represents the classical configuration
Coherentsuperposition Of Eigen states
• But specific structure of G(x) actually describes the behavior of the response to the classical configuration.
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• But specific structure of G(x) actually describes the behavior of the response to the classical configuration.
• Interaction with the environment is characterized by quantum entanglement:
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• But specific structure of G(x) actually describes the behavior of the response to the classical configuration.
• Interaction with the environment is characterized by quantum entanglement:
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Conditional /pointer state.
• But specific structure of G(x) actually describes the behavior of the response to the classical configuration.
• Interaction with the environment is characterized by quantum entanglement:
• Configuration space joint state vector:
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Conditional /pointer state.
• Reduced density matrix:
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• Reduced density matrix:
• Configuration space eigenstate satisfy:
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• Reduced density matrix:
• Configuration space eigenstate satisfy:
• Wave functional of joint system and environment:
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• Reduced density matrix:
• Configuration space eigenstate satisfy:
• Wave functional of joint system and environment:
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• Decoherence factor:
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• Decoherence factor:
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• Decoherence factor:
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• Wigner function:
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• Wigner function:
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Captures the effect of non-Gaussianity
