What I am going to outline should not be seen as a recommendation for most. Probably only a minority would benefit from it.
One possibility to learn the necessary maths to study physics, from a "mathematical physics" point of view is to first study a degree in maths (and maybe also a Master...
Usually it is stated that physics is divided among classical mechanics, classical field theory, quantum mechanics, quantum field theory and statistical mechanics, with hbar, the speed of light and the number of particles being the parameters differentiating all these theories.
However, despite...
I am a person working in the private sector. I studied physics, up to MSc level (QFT, string theory). But then I moved towards the private sector, raising a family, etc.
My wish is to try and understand QFT at the non-perturbative level. I do not need to write any paper on that subject, just to...
The ZFC axioms are statements combining "atomic formulas" such as "p ∈ A" and "A = B", using AND, OR, imply, NOT, for all and exists.
But (it seems to me, at least) there is the implicit assumption that the "atomic formulas", "p ∈ A" and "A = B", are considered to be propositions, i.e. they are...
In the past, I have asked in this forum about the concept of set membership, in the context of ZFC.
I guess it is a normal reaction to be a bit surprised by the usual statement in books that the set membership relationship is "undefined".
But I have had this idea: a typical definition of the...
Reading the interesting book "Groups_and_Manifolds__Lectures_for_Physicists_with_Examples_in_Mathematica", in the introduction it is stated:
(...) we have, within our contemporary physical paradigm, a rather simple and universal scheme of interpretation of the Fundamental Interactions and of...
In Quantum Mechanics, the position (or momentum) variable is quantized. I define "quantization" as promoting a variable into a probability distribution.
For example, with the double slit experiment, the classical assumption that the position/path of a particle is "unique" cannot explain...
In Euclidean geometry (and even in measure theory, see for example Stein and Shakarchi's Real Analysis), distance in the real numbers is defined as the difference of the real numbers, and area of a square is understood as the product of the distances defining the given square (and the same for...
I think the effective action should make sense also in Quantum Mechanics, not only in QFT. But I have never seen described in a QM book as such. Could there be a QM book that uses effective actions? Or maybe in QM effective actions are called another name?
I think effective actions in QM could...
The Schrödinger equation can be derived from the path integral quantization of the Lagrangian of classical, non-relativistic particles.
Can the Klein-Gordon (and maybe the Dirac) equation be derived from the path integral quantization of a given classical (supposedly relativistic) Lagrangian of...
I enjoyed a lot the three first volumes of Zeidler's planned series of 6 books on QFT. Unfortunately, he passed away too soon.
However, it is clear from reading the first three books, that an outline of the next books in the series was already planned.
Is there a draft, containing the basis of...
Wave optics, including diffraction, seems to be apt for path integral language. In fact, Feynman's double slit language is purely "diffraction". Also, the PDE for the wave equation results in a solution via Green's function, and the Green function is where "the path integral lives".
I have...
In analogy to classical mechanics, I thought a good definition to "What does "solving a quantum mechanics problem" mean?" was to give the propagator (aka the Green function, or the 2-point correlation function):
In classical mechanics, solving a problem means to give the path of the particle...
Goldrei's Propositional and Predicate Calculus states, in page 13:
"The countable union of countable sets is countable (...) This result is needed to prove our major result, the completeness theorem in Chapter 5. It depends on a principle called the axiom of choice."
In other words: the most...
Goldrei's Propositional and Predicate Calculus states (in my words; any mistake is mine) that first-order logic is complete, i.e. any logic deduction from a set of axioms (written in first-order logic) is equivalent to proving the theorem for all models satisfying the axioms.
Completeness is...
Of course, there is probability theory in QFT. The partition function can be understood as a characteristic functional.
What surprises me is there is little discussion about characteristic functionals, except for some 40 year old papers.
It seems surprising to me that physicists, being used to...
The Bohr atom gave the answer to the spectrum of the hydrogen atom.
But the spectra of stars contains many absorption (and sometimes emission) lines, corresponding to most atoms (up to iron, I believe).
And atmospheric absorption is also due to absorption of some molecules, such as water...
Let us imagine I want to create a podcast or a video, and I believe there is a book which is really spot on what I want to say.
Can I "inspire myself" in the book? Of course, in no circumstances I would use pictures of the book, or paragraphs from the book. All my words would be mine, so I...
Very often, the term "Green's function" is used more than "correlations" in QFT. For example, the notation:
$$<\Omega|T\{...\}|\Omega> =: <...>$$
appears in Schwartz's QFT book. And it seems very natural, basically because the path integral definition of those terms "looks like" the...
I know this question has been asked, in several ways, many times before. I have read many of the posts. And still I do not fully understand the situation: is QM in any way a subset of QFT?
Apparently no: QM uses position variables, while QFT does not. QM has the Born rule and a wave function...
Classical mechanics (and classical field theory) has the principle of stationary action (Hamilton's principle) as main principle. The Euler-Lagrange equations are derived from that principle, by using calculus of variations, on functionals (functions of functions).
Is there an equivalent...
The Feynman propagator:
$$D_{F}(x,y) = <0|T\{\phi_{0}(x) \phi_{0}(y)\}|0> $$
is the Green's function of the operator (except maybe for a constant):
$$ (\Box + m^2)$$
In other words:
$$ (\Box + m^2) D_{F}(x,y) = - i \hbar \delta^{4}(x-y)$$
My question is:
Which is the operator that...
I am trying to understand ergodic theory, i.e. how simple systems reach equilibrium.
I consider a classical particle in a 2D (or 3D) box. Funnily, I have never seen this example in books (probably due to lack of knowledge). Instead, in QM, the particle in a box is a prototypical example.
My...
I am in my forties, with a child and a firm to run (little to do with physics). I have a degree in physics, and a MSc in high energy physics (QFT, a bit of string theory, etc.). I did not continue with an academic career, though.
Now I feel I want to have, as a goal for the rest of my life, to...
I have been browsing this book, and it seems a quite interesting one. The traditional Statistical Mechanics is quite traditionally treated (so only average) but then, the linking of Statistical Mechanics with QFT, and the exact solutions in Conformal Field Theory, are quite nice.
But I do not...
I think the cluster decomposition states that products of space like separated observable decouple when sandwiched with states.
An analogy with statistical mechanics seems to suggest that we are stating there are no phase transitions. For example, in the Ising model all spins are correlated in...
From Weinberg, The Quantum Theory of Fields, Vol. 1, there is the statement that "the only way" to merge Lorentz invariance with the cluster decomposition property (a.k.a. locality) is through a field theory.
He uses this argument basically to justify that any quantum theory at low energies...
A not very well defined question:
Path integrals (and generalizations) are sums over configurations. A logical extension of that process would be to sum not over configurations, but over theories (configurations are possible solutions of a single theory).
Renormalization already plays around...
My main question is regarding whether the membership relation is taken as an undefined concept (as is usually hinted in set theory books) or if the membership relation can be defined within the language of first order predicate theory.
Let me describe a method to define the membership relation...
I have been trying to study first-order logic to have a sound basis on mathematical language. The main target is to have a clear path: I start with first-order logic (the language), then I go and study set theory, which is in fact a series of axioms (ie, a series of statements of the language)...