Questions tagged [mathematics]

DO NOT USE THIS TAG just because your question involves math! If your question is on simplification of a mathematical expression, please ask it at math.stackexchange.com The mathematics tag covers non-applied pure mathematical disciplines that are traditionally not part of the mathematical physics curriculum, such as, e.g., number theory, category theory, algebraic geometry, general topology, algebraic topology, etc.

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1answer
44 views

Exponent of tensor product of operators (context: weak measurement)

I'm stuck on some identity for an exponentiated tensor product of operators, $e^{\hat A\otimes\hat B}$. I'm learning weak measurement, reading the review by Kofman et al (2012, DOI: 10.1016/j.physrep....
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2answers
37 views

How to fit the data for the asymptotic limit?

Suppose we have a pretty good experiment data with very small random error. But our theory is limited, which makes the systematic error significant. Sometimes we do have a nice theory for the weak ...
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0answers
41 views

infinitesimal areas and large areas scalar or vector quantities? [duplicate]

Is it true that an infinitesimal area is a vector quantity and large area is a scalar quantity ?
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22 views

Measure theory and fractal measure

I want to study multifractal for turbulent but I need to learn measure theory for this. so I wanna introduce me some book for both?
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3answers
300 views

How to write down various metrics without coordinates?

For example Schwarzschild metric, or Alcubierre metric, but using only intrinsic (natural, canonical, etc.) physical objects (like length, angles, etc.) for relations between natural objects on ...
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1answer
21 views

A question on the transition matrix. (algebraic derivation)

I am trying to follow the derivation of the following identity: $$[\epsilon -H_0+i\eta]^{-1}T = [\epsilon - H +i\eta]^{-1}U$$ where $T$ is the transition matrix and $U$ is the potential caused by ...
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1answer
126 views

Is there an explanation for this unexpected similarity between binomial coefficients and waves?

Background Binomial coefficients appeal mostly in probability, combinatorics number theory etc so were were surprised when we observed something that appeared to belong more to physics than pure ...
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1answer
94 views

Could a reformulation of GR and QFT in terms of nonstandard analysis be useful?

Are there any serious reformulations of a quantum field theory of general relativity within a nonstandard analysis framework?  Is it possible that such reformulations (which are possible,  in ...
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3answers
87 views

Trouble with Math in Physics [duplicate]

I am a current high school student and I am very interested in physics, especially particle physics (that stuff is super cool!). Unfortunately, my school only teaches classical physics, so I have to ...
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2answers
46 views

How many linear combinations of harmonics or normal modes can describe the same periodic function as a Fourier series?

Please note that I am not asking how many terms in a linear combination can describe a specific periodic function but if given that there exist a set or linear combination of normal modes that ...
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2answers
125 views

Is $[A,\exp{B}]=0 \Rightarrow [A,B]=0$ true?

The backward direction is trivial and this one probably too, but I just can't find a convincing argument. $A$, $B$ are Operators on a Hilbert Space (Ket Space).
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A simple calculation in Peskin's and Schroeder's QFT book on page 608 chapter 18

I am trying to calculate the term: $$(t^a)_{ij} (t^a)_{kl}$$ In the book it's written that it equals to $$A\delta_{il}\delta_{kj}+B\delta_{ij}\delta_{kl}$$ and from using equation (18.40) $$tr[t^a](...
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1answer
35 views

Pseudotensors for describing physical quantities

I have been reading about tensors from Mathematical methods for Physics and Engineering, by K.F. Riley, M.P. Hobson and S.J. Bence. And there are a couple of things i am not getting. On page 949 (...
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1answer
147 views

About physical meaning of Hausdorff, Second Countable and Paracompact conditions of Manifold Theory

I would like to ask you, specially for the people here who deals with General Relativity/Differential Geometry, physical implications about Manifolds. Well, the most intuitive notion about a manifold ...
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2answers
91 views

What are change of frame and change of coordinates?

What's the difference between a change of frame and a change of coordinates? I feel like both are transformations on the coordinates but change of frame changes also the vectors.
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3answers
113 views

Reason behind vector addition law

What is the reason behind triangle law of vector addition, in other words, how is this really justified?
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1answer
33 views

Heuristic for large $x$ behavior from small $q$ behavior of Fourier Transform

If I have a function $h(\mathbf x)$ which may be written $$h(\mathbf x)= \int \frac{\text{d}^d\mathbf q}{(2\pi)^d} \, h(\mathbf q) e^{-i \mathbf q \cdot \mathbf x}$$ and assume spherical ...
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0answers
81 views

Deriving the stress-energy tensor from the Einstein-Hilbert action

I'm a mathematician who knows very little physics and is trying to learn relativity theory from a mathematical perspective. Let $M$ be a compact, orientable manifold. In the vacuum, the Einstein-...
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1answer
42 views

Propagation of Uncertainty vs Dividing Uncertainties

I have a quick question! When I’m calculating the uncertainty of a formula, like $v = d/t$ . . . What method do I use? Sticking with my example ($v = d/t$). Do I convert the uncertainties of ...
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2answers
135 views

Why cube roots or in general $(2n-1)$th roots are rarely seen in equations in physics?

I rarely saw any equation in physics which involved cube roots or odd roots.Even while solving problems I rarely saw any odd root or cube root. So why nature prefers even powers of physical ...
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1answer
50 views

Why is it convention to let vectors equal each other [closed]

I understand that the magnitudes of the two velocities equal each other. But I don't understand why it is more correct to use the convention. As $\vec v_1- \vec v_2$ should not equal $0$ it should $2 \...
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3answers
349 views

Why choosing for prime numbers eliminates vibration?

I have read that the spokes of a car wheel are usually five because, besides other substantial reasons, five being a prime number helps to reduce vibrations. The same also happens with the numbers of ...
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4answers
104 views

Is my understanding of vectors correct?

I recently learned that a vector in mathematics (an element of vector space) is not necessarily a vector in physics. In physics, we also need that the components of the vector on a coordinate ...
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2answers
58 views

Why do we choose the operator or supremum norm while proving unboundedness of the momentum operator?

In most sources, I've noticed that while proving the unboundedness of the momentum operator $\left(-i\hbar \frac{\partial}{\partial x}\right)$ the operator norm (or supremum norm) $\lVert\ .\rVert_\...
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1answer
76 views

Distance from ground to start suicide burn with initial height [closed]

In the figure below, the rocket is dropped with no initial velocity at a height of $h$. $d_f$ (free fall distance) is the distance in which the rocket is in free fall. $v_b$ is the velocity due to ...
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3answers
152 views

Why do my books introduce the equation $\nabla \cdot \mathbf{E}=\frac{\rho}{\epsilon_0}$ without showing partial derivatives of $\mathbf{E}$ exist?

In electromagnetism (electrostatics), we often come across the equation $\nabla \cdot \mathbf{E}=\frac{\rho}{\epsilon_0}$. In order for this equation to be meaningful, $\mathbf{E}$ must be a ...
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0answers
25 views

Lagrange Multiplier with inequality conditions

How can Lagrange multiplier method be used when inequality conditions are given instead of equality conditions?
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0answers
19 views

Mathematical bases/prerequisites for Feynman's “QED” [duplicate]

I have enjoyed studying Feynman's book "QED". I have sufficient mathematical maturity to go beyond his analogies. I am hoping someone has provided notes on the book enumerating the mathematical basis ...
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7answers
8k views

Why does Taylor’s series “work”?

I am an undergraduate Physics student completing my first year shortly. The following question is based on the physical systems I’ve encountered so far. (We mostly did Newtonian mechanics.) In all of ...
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1answer
28 views

Why is force 0 either side of an inflexion point in neutral equilibrium?

In Tipler & Mosca 5th edition p173 it defines neutral equilibrium as a point in a U-x curve where $\frac{dU}{dx}=0$ and also $\frac{dU}{dx}=0$ for a small displacement either side of the point. ...
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11answers
1k views

Is there a proof that the set of real numbers can exactly represent distances? [duplicate]

Mathematicians define real numbers in an abstract way - as an 'ordered field' with 'the least upper bound property'. In physics, we use real numbers to represent distances. For us to be able to do ...
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3answers
72 views

How to derive kinematics equations using calculus? [closed]

I read derivation of kinematics equations using calculus: $$a=\frac{\text dv}{\text dt}$$ $$\implies \text dv=a\text dt$$ $$\implies \int_{v_0}^v\text dv=\int_0^t a\text dt$$ $$\implies v-v_0=at$$ $$\...
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1answer
43 views

Derivative of tensor product of quantum states

Recently I asked a question over at the math stack exchange: https://math.stackexchange.com/q/3210375/. However I figured I'd ask here too, seeing as the question originated in a physics course I'm ...
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2answers
93 views

Infinite sum: Renormalisation

Trying to do the calculation made in a physics article Real-time Feynman path integral with Picard--Lefschetz theory and its applications to quantum tunneling (page 10 to go from equation 56 to 57), I ...
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0answers
79 views

Characteristics of the Navier-Stokes equations as a set of PDE's

I am not entirely sure if I should ask this question here or not, but here goes: can anyone suggest any reference (book, article, etc.) about the Navier-Stokes equations from a mathematical point view?...
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0answers
20 views

Which books or content should I prefer to understand particle physics operator and representations [duplicate]

I am just a intermediate passout and I want to know which books should I prefer for learning the mathematics of particle physics level like the different representation, operation like Hamiltonian, ...
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1answer
50 views

Is curvature the exterior covariant derivative of the connection?

Let $P\to M$ be a $G$-principal bundle, $G$ a topological group, $\omega$ the connection and $V$ a vector space. We define $d_\omega: \Omega^k_G(P, V)\to\Omega^{k+1}_G(P, V)$ the exterior covariant ...
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2answers
117 views

Precise definition of the Hilbert space in QM?

In QM books (at least those I have read) the definition of the Hilbert space used is somewhat blurred (the "space of square integrable functions" is not enough to define it precisely : which kind of ...
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2answers
307 views

Does it make sense to speak in a total derivative of a functional? Part I

I would like to consider the problem of the total derivative of a given functional \begin{equation} \mathcal{L}\bigg[\phi\big(x,y,z,t\big),\frac{\partial{\phi}}{\partial{x}}\big(x,y,z,t\big),\frac{\...
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1answer
81 views

Dirac delta function mathematical expression proof

In a discussion of the second order transitions in graphene this mathematical expression is used. $$ \left|\frac{1}{\varepsilon + i \Gamma/2}\right|^2 = \frac{2\pi}{\Gamma}\delta(\epsilon) $$ And I'm ...
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1answer
107 views

Is math truly infinite or is it just theory? [closed]

I imagine you can theoretically add +1 to any number But untill it happens it isnt so. That + no free will = math is limited and has a definite value. I imagine its tied in with conservation laws... ...
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41 views

How to prove that the limit of surface integral exists?

My book "Electromagnetic Fields" says in $\text{Section}\ 3.4$: Question Why does the limit in equation $(3.42)$ exist (convergent)? Why is the contribution from $(S-S_{\delta})$ remains ...
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44 views

A. Zee Contour Integral

In A.Zee's book I have come a cross an integral which I found difficult to solve.
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2answers
86 views

What are the theorems that constitute the Maxwell's equation? [closed]

Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
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2answers
120 views

Non-rigorous math books [duplicate]

I’m self-studying physics and mathematics out of interest and I am looking for some non-rigorous (text)books on mathematics. Perhaps one book covering all areas relevant to physics or separate books ...
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1answer
81 views

Diffusion equation with time-dependent boundary condition

I was trying to solve this 1D diffusion problem \begin{equation} \dfrac{\partial^2 T}{\partial \xi^2} = \dfrac{1}{\kappa_S}\dfrac{\partial T}{\partial t}\, , \label{eq_diff_xi} \end{equation} with ...
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1answer
75 views

Do non-abelian group mathematics have any use in the real world math? [closed]

I know physics uses a lot of non-abelian mathematics (though I cannot wrap my head around ab does not equal ba).. Is there any real world (macro world we live in) uses for non-abelian mathematics? ...
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35 views

Is there an easier guide or something to George Arfken's mathematical methods for physicists? [duplicate]

I'm doing bachelor's in Physics and in 3rd semester the professor introduced Mathematical methods with us, which seemed like a great book at first but by the time we started a chapter on Tensor ...
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2answers
151 views

Complex conjugated representation and its Young tableaux

This post is an exact copy of one that I posted in Math's site. I do this copy because people there suggested me to do it since, apparentely, in Mathematics and Physics we use different conventions ...
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0answers
99 views

Euler-Maclaurin formula for path integral

Is there a corresponding Euler-Maclaurin formula for path integral when we divide the path integral into discrete lattice? What is the error correction when we divide the space into lattice of length ...