Gravitational quantum mechanics: a theory for explaining spacetime. This a seminar on several scientific papers about quantum gravity Phenomenology which has been gathered several important outcomes.

1. Quantum Gravity!
A theory to explain spacetime
Milad Hajebrahimi
1

2. Content
The Quest for Quantum Gravity, Phenomenology
Examples of Phenomenological Models for Quantum
Gravity
Minimal Length & Maximal Momentum, Minimal
Momentum & Maximal Length
2

3. Content
Some Fundamental Features of this Gravitational Quantum Mechanics
Kernel Functions
• Particle-like Approach
• Wave-like Approach
Generalized Feynman Path Integrals
• Particle-like Approach
• Wave-like Approach
Generalized Feynman Path Integral and Some Thermodynamical Properties of an Ideal Gas
3

4. The Quest for Quantum
Gravity, Phenomenology
An introduction
4

5. The Quest for Quantum Gravity, Phenomenology
 General theory of relativity
 The standard model of particle
physics
 General relativity has so far refused
to be quantized!
 An entirely classical theory
 The result of its quantizing is non-
renormalizable
 There are three main reasons why
the present status requires a
solutions:
 Superposition states
 Singularities
 The black hole information loss
problem
5
This does not mean that one necessarily
obtains quantum gravity by quantizing
gravity.

6. The Quest for Quantum Gravity, Phenomenology
 Quantum particles can exist
in superposition states.
• These particles
respectively, carry energy
and thus gravitate.
• What their gravitational field
is: as a classical field it does
not exist in superpositions.
6
We don’t know what is the
gravitational field of a quantum
superposition.

7. The Quest for Quantum Gravity, Phenomenology
 General relativity predicts
the formation of
singularities.
• Infinite energy density and
gravitational forces
• Unphysical and signal a
breakdown of the theory
• Requiring a more
fundamental theory!
7
General relativity predicts its
own breakdown: Singularities

8. The Quest for Quantum Gravity, Phenomenology
 The black hole information loss
problem
• Black holes emit thermal radiation
• Any distribution with the same initial
mass that collapsed to a black hole
would eventually be converted into
the same thermal final state
• Detailed information contained in the
initial configuration would have
gotten lost
• Incompatible with quantum
mechanics
8
Black holes seem to destroy
information, and we don’t know how
that is compatible with quantum
mechanics.

9. The Quest for Quantum Gravity, Phenomenology
 Planck scale
 The scale at which effects of
quantum gravity are expected to
become relevant
 Energy, length and time
 To estimate the Planck scale:
 An amount of energy, 𝐸, in a
volume of size ∆𝑥3
 Via Einstein’s field equations:
 Consider the energy to be localized
as good as quantum mechanics
possibly allows us
 To its Compton wavelength:
 This distortion will become non-
negligible when
9

10. The Quest for Quantum Gravity, Phenomenology
This mass scale, which
corresponds to the Planck mass
and the related Compton
wavelength, the Planck length:
10

11. The Quest for Quantum Gravity, Phenomenology
Quantum gravity: any approach
that is able to resolve the
apparent tension between
general relativity and quantum
field theories, and to address
the three problems mentioned
above.
11
We will refer as ‘quantum gravity’ to any
attempted solution of these problems.

12. Phenomenological Models
 A phenomenological model:
 An extension of a known theory
that allows one to compute effects
of specific additional features
 Bridges the gap between theory
and experiment
 It guides the development of the
theory by allowing to test general
properties
 Not itself a fundamental theory
and therefore not entirely
self-contained
12
The first requirement for a good
phenomenological model is of course that it
be in agreement with already available data
and that it be internally consistent.

13. Examples of
Phenomenological
Models for Quantum
Gravity
None of them has made contact to experiment
13

14. Examples of Phenomenological Models for Quantum Gravity
Violations of Lorentz Invariance
 Does the ground state of spacetime
obey Lorentz-invariance?
 If Lorentz-invariance was broken,
observer-independence would be
explicitly violated and a preferred
frame would be singled out.
 The matter sector of the standard
model or the gravitational sector
 In the purely gravitational sector:
by coupling a tensor field, or its
derivatives respectively, to the
metric or the curvature
 Einstein-Aether theory
 In the standard model sector:
additional terms to the standard
model Lagrangian breaking the
symmetry
 Modified dispersion relations
14

15. Examples of Phenomenological Models for Quantum Gravity
Deformations of Special Relativity
 The possibility that special relativity
may be modified in the high energy
regime
 The modified Lorentz-
transformations in momentum
space have two invariants: the
speed of light and the Planck mass
 Results:
 Generalized uncertainty
 Curved momentum space
 Modified dispersion relation
 An energy-dependent speed of
light
15

16. Examples of Phenomenological Models for Quantum Gravity
Minimal Length and Generalized
Uncertainty
 There are several thought
experiments for probing smallest
distances that lead one to conclude
the non-negligible perturbations of
the background geometry close by
the Planck scale
 Incorporating a minimal length into
quantum mechanics and quantum
field theory results in:
 Generalized uncertainty principle
 Prevents an arbitrarily good
localization in position space
 Modified dispersion relation
 Modified measure in momentum
space
 Can be understood as a non-trivial
geometry of momentum space
 May or may not have an energy-
dependent speed of light
 Break Lorentz-invariance explicitly
16

17. Examples of Phenomenological Models for Quantum Gravity
 Causal Sets
 Space-time foam and granularity
 Geometrogenesis
 Loop Quantum Cosmology
 String Cosmology
 The Arkani-Hamed-Dimopoulos-Dvali Model
 The Randall-Sundrum Model
17

18. Minimal Length &
Maximal Momentum,
Minimal Momentum &
Maximal Length
Four Natural Cutoffs
18

19. 19

20. Minimal Length & Maximal Momentum, Minimal
Momentum & Maximal Length
Minimal Length
 Heisenberg uncertainty principle:
 From the Heisenberg’s Electron
Microscope Gedanken Experiment
 But he had disregarded the
gravitational interaction of the photons
with electron
 Gravity is coupled to everything
20

21. Minimal Length & Maximal Momentum, Minimal
Momentum & Maximal Length
Minimal Length
 The more carefully probing electron position the more
energetic photons so, the more gravitational interaction
 Planck scale
 So, an extra position uncertainty due to the gravitational
contribution of very high energy beams:
 Then HUP turns into GUP:
21
A nontrivial assumption:
The minimal length = A nonzero
uncertainty in position measurement

22. Minimal Length & Maximal Momentum, Minimal
Momentum & Maximal Length
Minimal Length
 The common form of GUP in the
presence of minimal length:
 The modified Heisenberg algebra:
22

23. Minimal Length & Maximal Momentum, Minimal
Momentum & Maximal Length
Minimal Length & Minimal
Momentum
 The existence of minimum position
and minimum momentum
uncertainty leads to:
 And a phase space commutator in
one dimension of the form:
23

24. Minimal Length & Maximal Momentum, Minimal
Momentum & Maximal Length
Minimal Length & Maximal Momentum
 The modified Heisenberg algebra in DSR theories:
 By asymptotic expression to first order:
 Corresponding Heisenberg algebra for this type of GUP:
24
A minimal measurable
length leads to a maximum
measurable momentum of
the order of the Planck
momentum which is an
upper bound on test particle
momentum

25. Minimal Length & Maximal Momentum, Minimal
Momentum & Maximal Length
Explicit Maxima and Minima in both
Position and Momentum Uncertainties
 A GUP of the form
25

26. Some Fundamental
Features of a
Gravitational Quantum
Mechanics
Some basic notions of a gravitational quantum
mechanics
26

27. Some Fundamental Features of a Gravitational Quantum
Mechanics
 Define the operators P and X:
 The generalized identity operator:
 The generalization of the scalar
product of momentum eigenstates:
 A maximal momentum cutoff
 the bounds of integrals over p
change from to
 Position space representation
should be modified
 Quasi-position space:
27

28. Some Fundamental Features of a Gravitational Quantum
Mechanics
 Standard quantum mechanics
 Absolute localization = Absolute
accuracy in the position
measurements
 Commutative spacetime
 With effects of gravity in the
ultraviolet regime
 Maximally localized up to the
minimal length
 Non-commutative structure of
spacetime:
 We can not build a Hilbert space
on the position space wave
functions
because there is no longer a zero
uncertainty in position
 There can not be any physical state
as a position eigenstate
 Ordinary position space
representation is no longer
applicable
28

29. Some Fundamental Features of a Gravitational Quantum
Mechanics
 Quasi-position space representation
 Maximal localization states
 Localizability limited to the minimum measurable distance
 Momentum space wave functions of the states that are maximally localized:
29

30. Some Fundamental Features of a Gravitational Quantum
Mechanics
 By projecting arbitrary states on maximally localized states, we can define the
state’s quasi-position wave function
 probability amplitude for the particle being maximally localized around x
 The transformation of a momentum space wave function into a quasi-position
space wave function:
30

31. Some Fundamental Features of a Gravitational Quantum
Mechanics
 the transformation of a quasi-position space wave function into a momentum
space wave function:
 Modified wave number in quasi-position space:
31

32. Some Fundamental Features of a Gravitational Quantum
Mechanics
 Modified wave length in quasi-position space:
 From the de Broglie relation:
 Modified form of the frequency in the quasi-position space
32

33. Some Fundamental Features of a Gravitational Quantum
Mechanics
 Modified Planck relation in quasi-position space:
 Quasi-energy
 Responsible for the time evolution of quasi-position wave functions
 Time-dependent wave function in quasi-position space evolves as
33

34. Kernel Functions
Propagators in the context of gravitational quantum
mechanics
34

35. Kernel Functions
 A general definition of kernel
functions K
 Standard quantum mechanics, a
definition of propagator:
 Probability amplitude to find a
point particle in the position 𝑥𝑓 at
the time 𝑡𝑓, while initially it was in
the position 𝑥𝑖 at the time 𝑡𝑖
 In the quasi-position formalism:
 Generalized form of propagator in
the gravitational quantum
mechanics
 Using identity operator:
35

36. Kernel Functions
 The modified Kernel function depends on the form of energy 𝐸 𝑝
 Particle-like
 Wave-like
 It is possible to construct a path integral even in the wave-like approach due to
the presence of natural cutoffs
36

37. Kernel Functions: Particle-like Approach
 With the modified form of the energy inspired by the quasi-momentum
 By some calculations (!!!)
37

38. Kernel Functions: Wave-like Approach
 With the modified form of the Planck relation:
 An explicit relation for wave-like kernel function due to existence of these cutoffs
38

39. Generalized Feynman
Path Integrals
Feynman path integrals in the language of
gravitational quantum mechanics
39

40. Generalized Feynman Path Integrals
 The kernel function K can provide a new formulation which was presented by
Feynman; the path integral formalism:
40

41. Generalized Feynman Path Integrals
 Modifications of Feynman path integral in the Planck scale
 By using the maximally localized states:
 We see that each expression is of the form:
 Again we have particle-like and wave-like approaches
41

42. Generalized Feynman Path Integrals: Particle-like Approach
 By using of the generalized energy relation with respect to quasi-momentum:
 An important outcome:
42

43. Generalized Feynman Path Integrals: Particle-like Approach
 Generalized form of the Feynman path integral in the presence of these cutoffs in
a gravitational quantum mechanics:
43

44. Generalized Feynman Path Integrals: Wave-like Approach
 By using of the generalized energy relation with respect to quasi-energy:
 Feynman path integral of a free particle in the wavelike approach where
quantum gravitational effects
44

45. Generalized Feynman
Path Integral and Some
Thermodynamical
Properties of an Ideal
Gas
Using of generalized path integral to modify
Thermodynamical parameter
45

46. Generalized Feynman Path Integral and Some
Thermodynamical Properties of an Ideal Gas
 Partition function of n-particle systems in position space
 The connection between the Euclidean path integral and statistical mechanics
 Replacing the time variable t with the Euclidean time τ
 A modified partition function from a modified path integral:
46

47. Generalized Feynman Path Integral and Some
Thermodynamical Properties of an Ideal Gas
 The modified partition function for a bosonic ideal gas N particles:
47

48. Helmholtz Free Energy
 Helmholtz free energy:
48

49. Chemical Potential
 Chemical Potential:
49
The solid thick curve
represents the
chemical potential in
the classical
theory

50. Entropy
 Using the expression for the free energy:
 Entropy of an ideal gas in the presence of natural cutoffs:
50

51. Specific Heat
 Specific heat in a constant volume:
51
The solid thick curve
represents the
chemical potential in
the classical
theory

52. Internal Energy
 Having the expressions for the free energy and entropy:
 Internal energy U in the presence of natural cutoffs:
 The value of internal energy of an ideal gas is not divergent
52

53. 53
Cassini, NASA