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1

All these functions (and more) are documented in the Handbook of mathematical functions by Abramowitz and Stegun (which is a staple of many professor libraries)


0

There are no proofs in physics. PERIOD FULL STOP. There are only measurements and models that are fit numerically. If that is not what you are looking for, you will have to stick with mathematics.


1

Both $L^p$-spaces and Sobolev spaces are actually defined via equivalence classes (this is at least one of many equivalent ways). By definition, two functions that do not differ in norm are representants of the same function as already explained. Now, note that the Sobolev norm is just the sum of the $L^p$ norms for weak derivatives (to whichever order you ...


1

However then I say: Why do you make things so complicated? Suppose you want to calculate $\exp(A)$. Why don't you define $$\exp(A)~:=~1+A+1/2 A^2 + \ldots $$ and require convergence with respect to the operator norm. An example: Consider the vectorspace spanned by the monomials $1,x,x^2,\ldots$ and let $A=d/dx$. Then you can perfectly define ...


1

I think expecting "fluid" behaviour in terms of a material that does not support shear, is not useful in the context of the various systems you have listed in the question. Instead, I believe you are intuitively connecting ideas and concepts pertaining to conservation laws. So the idea that in specific systems, conserved charges (in the sense of Noether) ...


2

Incompleteness of a coordinate system is not a canonical definition as that, for instance, of geodesical (in)completeness. It simply means that the domain of the coordinate system does not cover the whole manifold (and perhaps there are several inequivalent extensions of the initial manifold represented by the given domain of the coordinate system). If a ...


2

Typically, the Hilbert spaces one considers in quantum mechanics are $L^2$ spaces. The elements of these spaces are equivalence classes of functions which differ only on a null set of points, i.e. whose distance in terms of the $L^2$ norm is zero, $\|n-\tilde n\|_{L^2}=0$. That is, you are right, but it's the $L^2$ norm that matters, not the Sobolev norm. ...


1

Here is a proof following Ojima, "Lorentz Invariance vs. Temperature in QFT", Letters in Mathematical Physics (1986) Vol. 11, Issue 1 (1986) 73-80. The first two pages of the paper are available for free here, but the website wants money for more of the paper. (Click the orange "Look Inside" button if the paper doesn't open automatically.) Fortunately, the ...


3

Ghostly Lie algebra cohomology Let $\mathfrak{g}$ be our Lie algebra and $V_\rho$ a representation space with representation map $\rho : \mathfrak{g} \to \mathrm{End}(V_\rho)$. $V_\rho$ is, by the action through the representation, naturally a $\mathfrak{g}$-module (people missing the ring structure in $\mathfrak{g}$ - just embed it into the universal ...


3

This is a typical case of a problem which is clear enough physically speaking, but mathematically messy. Where rigorous results are folkloristically employed to achieve some result which, actually, would need much more care in deriving it... But presumably, mathematical details would not change the physical picture. Here the difference between theoretical ...


1

The uniqueness theorem actually stems from differential equation mathematics. If you have a complete set of 1) Diff eq. 2) Boundary conditions Then you have a single solution. This means also that if you found a solution that fulfils these conditions, it is the only solution you have. In the so called mirror problem what you ...


2

The assumption that $\mathcal{D}$ is invariant under $\phi(f)$ for each $f\in \mathcal{S}(\mathbb{R^4})$, the Schwartz space of functions of rapid decrease is one of the Wightman axioms. Its main use is for the vacuum expextation values $(\psi_0,\phi(f_1)...\phi(f_n)\psi_0)$ to make sense (where $\psi_0$ is the vacuum state), what would not happen in general ...


0

There is another reason why such a potential needs further attention. The point is that the system must have a unitary time evolution which is generated by a self-adjoint Hamiltonian. Thus one has to verify that $H={-{(∂/∂_{\bf{x}})}^2}/(2m)+V(x)$ defined on a suitable dense domain (for instance smooth square integrable functions vanishing in a neighbourhood ...


0

the ground state energy cannot be bounded from below and system is not stable/possible in quantum mechanical sense. in fact, the system could radiate an infinite amount of energy (photons) which for example would make a fundamental law such as the conservation of energy useless. So, is the following implication true? Force is not possible in QM → ...


1

This is a very difficult question. In one dimensions a two body scattering is the minimum interaction you can have. In multiple dimensions you can have more bodies scattering as the minimum interaction. The Yang Baxter equation is formally, $$(R\otimes \mathbf{1})(\mathbf{1}\otimes R)(R\otimes \mathbf{1}) =(\mathbf{1}\otimes R)(R\otimes ...


-1

The Hilbert space of an electron is the electron itself ! Even in classical mechanics we can replace the subjective notion of a system by your own phase space, this is not different in quantum mechanics. The phase space in classical mechanics tells us how to build physical quantities: as a function defined in the phase space to the real line. In quantum ...


0

In the following all integrals are done over the whole real line. The only one I can give an in depth explanation for, in terms of calculation, is the free particle. $$U(x,t;x_0) = <x|U(t)|x_0> = \int dp <x|e^{-ip^2/2m\hbar}|P><P|x_0>$$ where the $P$ is a momentum eigenstate and the Dirac notation indicates a jump from $x_0$ into $P$ ...


1

To calculate explicitly the curvature and geodesic equations for the conical spacetime you need an explicit metric. The metric $ds^{2}=dr^{2}+r^{2}d\phi^{2}$ describe a conical spacetime in the range of definition of the coordinates $(r,\phi)\in (0,\infty)\times [0,2\pi-\alpha)$. You can notice that this metric describe a flat spacetime in the domain of ...


1

It is a distinction corresponding to different types of spectral measures. The absolutely continuous spectrum corresponds to absolutely continuous measures, singular spectrum to continuous singular measures (both with respect to Lebesgue measure). Refer e.g. to Reed-Simon Chapter VII for a more detailed description.


6

OP considers the 'same-time' functional derivative (FD) $$\tag{1} \frac{\delta f(t)}{\delta x(t)}~:=~\frac{\partial f(t)}{\partial x(t)} - \frac{d}{dt} \frac{\partial f(t)}{\partial \dot{x}(t)} +\ldots. $$ Here $f(t)$ is shorthand for the function $f(x(t), \dot{x}(t), \ldots;t)$. Although the 'same-time' FD (1) can be notationally useful, it has various ...


5

In general functional derivatives obey chain and product rules. If the concept troubles you you can always think of a function as a vector with an infinity of coordinates. Then a functional derivative is just a partial derivative. If $F[h]$ is a functional of the function $h(x)$. You can think of this as $$ h \to \vec{h} = \left(h(x_1), h(x_2), ..., ...


4

Yes. Here, we are dealing with functional derivatives and these satisfy the chain rule and the product rule, which is really an important reason why it can be called a derivative to begin with. Important note: The definition that you give for the functional derivative is not the standard one, and does not satisfy its usual properties (as shown by ...


0

In circuit analysis we make some assumptions and we use shorthand notations frequently. For example we assume the potential doesn't vary anywhere in the wire, even it was 1 km long. Hence we don't write the spatial coordinates like we do in electromagnetics, in that they complicate the analysis and don't give much accuracy (the dimensions of the wire and ...


3

The electric potential is always a function of both spatial position and time, in both circuit theory and electrodynamics. $V$ is constant along a wire, so in circuit analysis you can take a short cut by specifying the position by just specifying which wire you're talking about, like $V_{1}(t)$, instead of specifying three spatial coordinates. $V$ doesn't ...


1

When you consider a voltage $V(t)$ in a circuit, you are talking about a voltage at a specific point in the circuit - in other words, implicitly it's $V(t,x,y,z)$ - at a certain time & place. When you have a static situation (things don't change over time) it's possible to talk about a potential as a function of location: $V(x,y,z)$ describing the ...


1

What you need is Kirchhoff's Matrix-Tree Theorem which expresses ${\rm det}\ A$ as a sum of trees. You can find an easy "Fermionic" proof of this theorem and a list of original references in my article "The Grassmann-Berezin calculus and theorems of the matrix-tree type" (arXiv version here if you do not have access to the journal).


1

I think you should have a look at Reed&Simons classic book on mathematical physics ("Methods of Modern Mathematical Physics", 4 volumes). Excellent and clear writing style, many further references and it covers most of the important analytic methods which are used in physics. For geometry stuff, I recommend Bishops "Tensor analysis on manifolds". Very ...


1

For an introductory book on the topic, consider "Numerical and Analytical Methods for Scientists and Engineers, Using Mathematica" by Daniel Dubin (ISBN-13: 978-0471266105). Whats sets this book apart from the rest is that it combines theoretical physics, teaches the math, and solves practical physics problems both by hand and by using Mathematica. This ...


2

Hints: Assume that $H$ is a complex Hilbert space. Assume that $A:H\to H$ is a normal operator$^1$. Then a version of the Spectral Theorem says that $A$ is orthonormally diagonalizable. Let $(\lambda_i)_{i\in I}$ denote the set of different eigenvalues of $A$ with corresponding multiplicities $(m_i)_{i\in I}$. Let $P_i$ be the orthogonal projection ...


2

The fine structure constant is given as: $$\alpha = \frac {k_{e} e^2} {\hbar c^2}$$ Immediately we have a problem in determining the rationality or otherwise of $\alpha$. The Coloumb constant, Planck constant (maybe not?) and speed of light are all either exact numbers or pre-defined. Since the elementary charge $e$ is an empirically derived constant we can ...


3

This is a situation where knowing the history of the terminology can be helpful. The QFT/string theory terminology comes from algebraic geometry, where the term moduli space is used for any space whose points correspond to some kind of geometric object. The projective space $\mathbb{P}(V)$, for example, is the moduli space of lines in the vector space $V$. ...


8

I) It is worthwhile mentioning that there exists a basic approach well-suited to physics applications (where we usually assume locality) that avoids multiplying two distributions together. The idea is that the two inputs $F$ and $G$ in the Poisson bracket (PB) $$\tag{1}\{F,G\} ~=~ \int_M \!dx \left( \frac{\delta F}{\delta \phi(x)}\frac{\delta G}{\delta ...


1

What you are proposing is known as the Heisenberg picture of quantum mechanics which had a primitive formulation even before the Schrödinger formulation in the form of Matrix mechanics. The linked wikipedia article is very well written, so I think me giving a detailed description would be redundant. In this picture, all the evolution is transformed into ...


1

We are very well allowed to ask for exact values of $x$ and $p$. We can just measure them. What we can't expect is that if we reprepare the same state, we'll get the same answer, if we measure again. We can also measure the position and the momentum and will obtain a specific answer - once again, if we reprepare the same state, we'll not get the same ...


1

If I've understood you correctly, you want rigourous mathematical formalism to treat PDE solutions which are not differentiable or not square-integrable etc. That is, you can have those point charges with fields blown up on them, fields being not differentiable on boundaries and so on. There exists a rigourous formalism to treat such things. It is called ...


1

The stationary action principle and the Euler-Lagrange (EL) equations are very broad and general constructions. The field variables in the variational principle could in principle map into some generic manifold $M$. On the other hand, Euler-Poincare (EP) equations appear in the special situation where the manifold is a Lie group $M=G$, and the action is ...


1

Something like $\textbf{E}(\textbf{r}) \propto (\textbf{r}-\textbf{r}^{\prime})/|\textbf{r}-\textbf{r}^{\prime}|^{3}$ due to a static point charge does not belong to the appropriate class of functions (differentiable, square-integrable, etc.) for $\textbf{E}$ fields. In a mathematically precise sense, it is not a solution to Maxwell's equations. Still, we ...


2

One of the major issues that seems to be going on here is the notion of point and surface structures in our 3D world. When we define electrostatic fields by a distribution of point charges, we are being somewhat non-physical. If we keep zooming in on an electron, it's going to start not looking like a point charge anymore. Consider the Darwin Term in the ...


1

The relation $$\langle a|b\rangle\propto\delta(a-b)$$ is nothing unusual, it is simply an orthogonality condition. If the proportionality was an equality, and in addition we had completeness, the set of states would form an orthonormal basis. The reason why the delta function shows up is that you assume your operator to have a continuous spectrum of ...


1

The steps that you wrote down till eq. 1 is in fact a simple proof of the following theorem (which can be looked up in elementary text books of quantum mechanics): Eigen functions (of a Hermitian operator or more generally a symmetric operator on a separable Hilbert space) belonging to distinct eigenvalues are orthogonal. This is always true for separable ...


0

But these fields diverge/become infinite in case of point charges, how is this justified and mathematically consistent ? If the charge $q$ of the particle is finite, both densities have to be singular at the point where the particle is, because a regular (everywhere finite) density would need to be non-zero in a region of non-zero volume to give finite ...


2

While i was typing, two good answers were posted. Since I don't want to delete everything, I'll leave this here nontheless. Without appealing to Lie theory, one might argue by physical reasoning. The unitary operators your book has in mind depend on a continous parameter $\alpha$. They describe continous transformations of the quantum mechanical state ...


3

Well, quantum mechanics is famous for not being intuitive for earthlings like us, but the following couple of facts might help: Observables in quantum mechanics are Hermitian/selfadjoint operators. The spectrum ${\rm Spec}(\hat{A}) \subseteq \mathbb{R}$ of a Hermitian/self-adjoint operator $\hat{A}$ belongs to the real axis $\mathbb{R}\subseteq ...


11

There's no escaping the Lie theory if you want to understand what is going on mathematically. I'll try to provide some intuitive pictures for what is going on in footnotes, I'm not sure if it will be what you are looking for, though: On any (finite-dimensional, for simplicity) vector space, the group of unitary operators is the Lie group $\mathrm{U}(N)$, ...


1

Hint: Establish first that $$\delta(xy)~=~\frac{\delta(y)}{|x|}+\frac{\delta(x)}{|y|}. $$


1

Let there be given an $n$-dimensional manifold $(M,\nabla)$ endowed with a connection $\nabla$. [In particular, we do not assume that the manifold $M$ is equipped with a metric tensor.] Let there be given a curve $\gamma:\mathbb{R}\to M$. Here the reader should think of $\mathbb{R}$ and $M$ as time and space, respectively. If $f: M\times \mathbb{R}\to ...


11

We have also the same notions of derivation, curl, etc... for functions that are less regular. When you write Maxwell's equations, you are writing a system of partial differential equations. To investigate them, you have to specify the type of solution you look for (in the language of PDEs: classic, mild, weak...) and the functional space you set your ...



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