This post begins a thread about Quantum Field Theory (QFT) in curved spacetime (ST). They are based on some of my notes about the subject. Enjoy it!
I. LIMITS FROM QUANTUM MECHANICS AND GENERAL RELATIVITY.
The Einstein-Hilbert action in General Relativity (GR) reads
The Planck length is
Therefore, the classical GR action is proportional to
Quantum curvature will become important when , and quantum corrections will be important if
and the path integral will read
Semiclassical gravity approach treats gravity as a classical field and the other fields as pure quantum fields. Of course, it is only an effective theory or approximation to the realm of pure quantum gravity. However, it is well defined in the limit if .
The Heisenberg Uncertainty Principle (HUP) reads
Due to the energy-mass equivalente , then we write
Whenever , then , that is, when , Quantum Gravity becomes inevitable!
Heisenberg Uncertainty Principle is sometimes extended to the Spacetime Uncertainty Principle (non-commutative geometry based, pioneered by Snyder and others), or SUP:
There are some interesting extensions of the HUP and SUP I am not going to cover in this thread. However, let me remark that an interesting relationship and link between IR (infra-red, long distances) and UV (ultra-violet, short distances) or IR/UV entanglement/interplay enters here when we combine QFT or QM with GR:
Remember the notion of Schwarzschild radius, related to the scape velocity of light in a Black Hole (BH):
so, when we get
Collider tests allow us to test quantum dimensions with spectrum
II. INTERLUDE: BACKGROUND IN QM AND CLASSICAL MECHANICS.
Firstly, we can compare Quantum Mechanics (QM) with Classical Mechanics (CM). Any physical theory answers some basic questions:
1st. What are the states or degrees of freedome?
2nd. What can we observe or measure?
3rd. How does the system evolve in time?
CM answers in the following way:
1) States are generalized coordinates and generalized momenta , plus some initial conditions or boundary conditions.
2) Observers “see” functions of generalized coordinates and momenta , i.e., observers “see” the phase space!
3) Systems evolve with equations of motion (EOM) provided by some lagrangian or hamiltonian function (). The EOM plus initial (boundary) conditions are a well posed problem in lagrangian or hamiltonian dynamics (excepting some “degenerated” systems that deserve special attention).
By the other hand, QM answers a bit differently as follows:
1) States are “rays” in a complex (projective) Hilbert space, .
2) Observables “are” (by definition in standard quantum mechanics) self-adjoint operators having real spectra and acting on some Hilbert space. Since , then we have
3) Evolution follows according certain unitary operator via the Schrödinger’s equation
This is the Schrödinger picture. OR we can leave fixed the state and then operators evolve with the EOM according to the so-called Heisenberg picture, the analogue of the classical mechanics approach with Poisson brackets extended to quantum commutators.
The canonical approach to quantization is review now:
A) Take a lagrangian with a harmonic oscillator term with unit mass (m=1):
The EOM reads
The hamiltonian reads:
The canonical momentum is
B) Canonical equations:
C) Canonical commutators ():
and where are Hermite polynomials, and is a normalization constant. Indeed, the general state for this system is a linear superposition
Remember that for the harmonic oscillator.
D) Creation/Annihilation representation for the standard harmonic oscillator (SHO):
In terms of these operators, we have
III. PARTICLE CREATION IN CURVED SPACETIME.
The general idea of QFT in curved spacetime and the phenomenon of pair creation is the following. Consider some function or observable F(t) during some time interval . Then, we have
The consequences are the definition of two set of “states” (in, out) in the asymptotic limit:
are aymptotic states in the past and future where the notion of particle or more generally “quasiparticle” is well defined. In general, we also write
We define the scattering matrix (popularly known as S-matrix) as
It is generally a complex number (it has complex components!). Then, represents the effect of particle physics interactions between asymptotic states. In curved spacetimes, the curvature acts as certain external field .
See you in my next QFT post!