Exercises in Statistical Mechanics
... Reσ(ω) =
kB T 0
(c) Write the Diffusion constant D in terms of the velocity-velocity correlation function, assuming that this correlation has a finite range in time.
Use Kubo’s formula from (b) in the DC limit of zero frequency to derive the Einstein-Nernst formula for the
mobility µ = ne
= eD/kB ...
Problem set 7
... Problem set 7
Due by beginning of class on Monday Mar 5, 2012
BCH formula for x and p , SHO
1. Consider the function f (t) = etA Be−tA where A, B are a pair of operators (e.g. position and
momentum or creation and annihilation operators etc.). t is a parameter which could be a time
interval or a spa ...
Kant and Kantianism
... First Formula: Universal Law:
1. The Formula of Universal Law/Formula of
Universal Law of Nature:
– "Act only according to that maxim whereby you
can at the same time will that it should become a
– "Act as if the maxim of your action were to
become through your will a universal law ...
Planck`s Law and Light Quantum Hypothesis.
... that is, the relation between the radiation density and the mean energy of
an oscillator, and they make assumptions about the number of degrees of
freedom of the ether, which appear in the above formula (the first factor
on the right– hand side). This factor, however, can be derived only from
A path towards quantum gravity
... History of singularity theorems
• Penrose: singularities in gravitational collapse
• Hawking, Geroch, Hawking—Ellis, HAWKINGPENROSE, Kriele: spacetime singularities are
generic properties of GR, provided that energy
conditions hold. Very little use of the Einstein
equations is made. Key role of top ...
Relativity + Quantum + Gravity
... • These three concepts are the basis of physics.
• They contain the three fundamental constants
c,ħ,G , which form a complete system of units.
• Are these concepts compatible with each other?
- Relativity and gravity (= general relativity) are
designed to be compatible.
- Relativity and quantum phys ...
Quantum Mechanical Derivation of the Wallis Formula for $\ pi$
... was derived by John Wallis in 1655  (see also ) by a method of successive interpolations.
While several mathematical proofs of this formula have been put forth in the past (many just
in the last decade) using probability , combinatorics and probability , geometric means
, trigonometry ...
Program - LQG
... We will present a general mechanism for the emergence of an effective classical
spacetime from a fundamental theory of quantum cosmology coupled to matter. This
idea is based on QFT on quantum spacetime, and the emergent classical metric is not
just the naïve expectation value of a ``metric operator ...
... Asymptotic freedom: g(m) runs to 0 when m grows
GR is perturbatively non-renormalisable: no running gravity coupling G(m)
What is Entanglement? Entangled Fields Looking at Entangled
... cannot really be considered as separate objects, but
are intrinsically connected or “entangled” with each
other. For example, performing a measurement on
one, such as measuring the particles spin as in the
diagram above, instantaneously affects the result of
measuring the other. This phenomena occur ...
Abstracts of the talks
... the same genus need not be equivalent, but one can show that there are only finitely many
equivalence classes of quadratic forms within a genus. In fact, there is an explicit formula
for the number of equivalence classes of quadratic forms in the genus of a fixed quadratic
form q (counted with multi ...
TALK - ECM
... provided by all modes with wave number k' > k
The dynamics of the relevant mode is obtained by
tracing over the environment.
This generally leaves the relevant mode in a mixed
state, whose evolution is determined by a
Feynman-Vernon influence functional (IF)
Large N quantum system
... • Problem Infinite number of solutions.
• f like a Nambu-Goldstone boson.
• Fix: Remember that the symmetry is also explicitly
broken (like the pion mass).
Quantum Fields in Curved Spacetime
... • The next order subtracts the constant in D0: the contribution to
the effective action diverges at both small s and large s
• This term modifies the IR so it is not a local term in the IR.
• It is wrong to subtract it.
Clément Hongler Spring 2016 Lecture Series EPFL
... conjectural, major recent progress in the field of rigorous conformal invariance allows
one to make mathematical sense of much of the story: In particular, for the Ising model
we are close to reaching a complete understanding of how it works. The series will thus
focus on giving an intuitive underst ...
The Asymptotic Safety Scenario for Quantum Gravity Bachelor
... less than the energy needed to probe these effects. The standard approach of quantizing
gravity in a perturbative quantum field theoretical framework using Feynman graphs
and the spin two graviton as its force carrier is plagued by many non renormalizable
ultraviolet divergences. It is however possi ...
... are called mutually unbiased if and only if
a b 1/ d
" Quantum gravity": an oxymoron
... The innumerable and learned efforts during seventy years to create a
quantum formulation of general relativity have only beaten the air – et pour
cause, as we shall see. On the other hand, it is evident to any unprejudiced
scientist that definite reasons must be at the root of this failure.
First of ...
Bilbao - INFN - Sezione di Firenze
... framework into something predictive ... and testable.
Many unresolved puzzles in gravitation and cosmology
(big bang, black holes, cosm..) probably do need a
consistent way to combine GR and QM
Insisting on theoretical consistency has paid off
enormously towards understanding EW and Strong
... microscopic theories : where the laws are
effective theories : where observations are made
effective theory may involve different degrees of
freedom as compared to microscopic theory
example: microscopic theory only for fermionic
atoms , macroscopic theory involves bosonic
collective degr ...
Physics 120 Homework Set #1 (due Sunday
... 6) In the “Illusion of Gravity” article, the author states: “A quantum theory of
gravity will probably provide us with an entirely new perspective of what spacetime
is.” Also, “if matter obeys the laws of quantum mechanics and gravity obeys the
laws of general relativity, we end up with mathematical ...
... If you try to compactify away a space R -> 0 it returns as an infinite R’
Asymptotic safety in quantum gravity
Asymptotic safety (sometimes also referred to as nonperturbative renormalizability) is a concept in quantum field theory which aims at finding a consistent and predictive quantum theory of the gravitational field. Its key ingredient is a nontrivial fixed point of the theory's renormalization group flow which controls the behavior of the coupling constants in the ultraviolet (UV) regime and renders physical quantities safe from divergences. Although originally proposed by Steven Weinberg to find a theory of quantum gravity, the idea of a nontrivial fixed point providing a possible UV completion can be applied also to other field theories, in particular to perturbatively nonrenormalizable ones. In this respect, it is similar to Quantum triviality.The essence of asymptotic safety is the observation that nontrivial renormalization group fixed points can be used to generalize the procedure of perturbative renormalization. In an asymptotically safe theory the couplings do not need to be small or tend to zero in the high energy limit but rather tend to finite values: they approach a nontrivial UV fixed point. The running of the coupling constants, i.e. their scale dependence described by the renormalization group (RG), is thus special in its UV limit in the sense that all their dimensionless combinations remain finite. This suffices to avoid unphysical divergences, e.g. in scattering amplitudes. The requirement of a UV fixed point restricts the form of the bare action and the values of the bare coupling constants, which become predictions of the asymptotic safety program rather than inputs.As for gravity, the standard procedure of perturbative renormalization fails since Newton's constant, the relevant expansion parameter, has negative mass dimension rendering general relativity perturbatively nonrenormalizable. This has driven the search for nonperturbative frameworks describing quantum gravity, including asymptotic safety which — in contrast to other approaches—is characterized by its use of quantum field theory methods, without depending on perturbative techniques, however. At the present time, there is accumulating evidence for a fixed point suitable for asymptotic safety, while a rigorous proof of its existence is still lacking.