# Tag Info

25

Straight from the horse's mouth: Source: Bureau International des Poids et Mesures (Search for "dimensionless" for all guidelines.) The International Bureau of Weights and Measures (French: Bureau international des poids et mesures), is an international standards organisation, one of three such organisations established to maintain the ...

16

The symbol $\Delta$ refers to a finite variation or change of a quantity – by finite, I mean one that is not infinitely small. The symbols $d,\delta$ refer to infinitesimal variations or numerators and denominators of derivatives. The difference between $d$ and $\delta$ is that $dX$ is only used if $X$ without the $d$ is an actual quantity that may be ...

13

The main distinction you want to make is between the Green function and the kernel. (I prefer the terminology "Green function" without the 's. Imagine a different name, say, Feynman. People would definitely say the Feynman function, not the Feynman's function. But I digress...) Start with a differential operator, call it $L$. E.g., in the case of ...

9

Here is a video of the film's science advisor explaining what the equation is and how he came up with it: http://www.youtube.com/watch?v=WjfT6MqTCqQ It is based on the Gompertz equation, which is a model of mortality rates, with some added "mathematical glitter."

8

Ben-Zvi & Frenkel denote vertex algebras $V$,$W$,... They're using the labels specifically for the spaces of states, but one could also use them to refer the whole package. Alternately, one sometimes sees all caps abbreviations: $YM_2$, $SYM_{4,G}$,... There is not to my knowledge any conventional notation for morphisms of field theories.

8

Each of the indices in a tensor have a particular left-right ordering. This ordering cannot be changed unless the tensor has some particular symmetry that permits it (or rather, that equates different components on interchange). The up-down positions of indices tells us about whether the index is associated with using a basis vector (up) or a basis ...

7

There is, I think, no really standard symbol for the generic (chiral) CFT used universally, but there is within the different formalizations. When chiral CFTs are modeled by vertex operator algebras, the standard symbol is usually "$V$" (for obvious reasons) as user388027 notes in his reply.. When chiral CFTs are modeled as conformal nets, then (as you ...

7

It's an integral over a closed line (e.g. a circle), see line integral. In particular, it is used in complex analysis for contour integrals (i.e closed lines on a complex plane), see e.g. example pointed out by Lubos. Also, it is used in real space, e.g. in electromagnetism, in Faraday's law of induction (part of the Maxwell equations, written in an ...

7

Are those square brackets standard notation in Physics? Yes. See, for example Sean Carroll notes. At least I can tell you from two other classic references using that notation, "General Relativity" by Wald (1984) and "A First Introducion to General Relativity" by Schutz (2009 for the most recent edition)  If I am in a non-curved $\mathbb M$ ...

7

The antisymmetric part is defined as $$A_{[a_1 \cdots a_n]} = \frac{1}{n!} \sum\limits_{\sigma \in P(n)} \text{sgn}(\sigma)A_{a_{\sigma(1)} \cdots a_{\sigma(n)}}$$ where $P(n)$ is the set of all permutations of the set $\{1,\cdots,n\}$. $\text{sgn}(\sigma)$ is called the sign of the permutation and is positive of $\sigma$ is obtained from the identity ...

6

In component notation, 3d and 4d vectors are usually distinguished using latin and greek letters respecitively, e.g. $u_i$ and $u_\mu$. Moreover, four-vectors without indices are usually just written as $u$, whereas three-vectors are denoted $\vec u$, as you say. You'll hardly find $\vec u$ denoting a four-vector. The option $\underline{u}$ is also ...

6

My taste, never overload your notation unless its necessary. Many people in quantum information try to avoid "hats" or further ornaments for operators that are just linear maps. Simple capital letters are fine to write Hamiltonians, channels, unitaries and measurements (italics are not really important, but its a de-facto standard). When people write ...

6

For conformal nets $\mathcal A,\mathcal B,\ldots$ or $A,B,\ldots$ is typical. For Virasoro nets $\mathrm{Vir}_{c=\frac 12}$ is normally used and for loop group nets $\mathcal A_{G_k}$. In VOA it seems to be common to use $V$ for a generic VOA. Kac uses in "VOA for Beginners" $V_Q$ for the lattice VOA associated with a lattice $Q$ and $V^k(\mathfrak g)$ for ...

6

The convention I have seen in journal articles, and that I prefer, is to simply omit any mention of units for dimensionless quantities. EDIT: I also see the style Emilio Pisanty recommends, particularly in tables and graphs. For a graph, the idea is that the datapoints you are plotting are actually numbers, so you want to divide them by the relevant base ...

5

As you see, there are different notations for quantum mechanical. Typically, even within a journal there is no one typesetting (style guides usually don't touch this topic). Besides the ones you mentioned, sometimes people use: bold font (e.g. ${\mathbf H}$), small font for operators acting on subsystems. Try looking at common notations used by your ...

5

It's common to put a hat over anything that's an operator instead of a c-number, so that $\hat A$ is an operator, $A$ is a c-number. Then we can use any letter as either an operator or as a c-number. Your $\hat O$ or $O$ to some extent suggests something that is specifically an observable quantity, just as $\hat H$ suggests a Hamiltonian operator, although ...

5

The facts that there is a sum over $i$ but the product doesn't involve $i$; the product is a product of exponentials, which as a major result (boxed and marked "DO NOT ERASE") would typically be written as a single exponential; given that the first equation is a differential equation, one should expect the second equation that gives $\Phi$ to be either an ...

5

In the Einstein convention, pairs of equal indices to be summed over may appear at the same tensor. For example, the formula ${A_k}^k=tr~A$ is perfectly legitimate. But your formula looks strange, as one usually sums over a lower index and an upper index, whereas you sum over lower indices only, which doesn't make sense in differential geometry unless your ...

5

$M$ is reducing. Thus, $\mathrm dM$ has a negative value. In contrast, in the above equations, you can see an $M-\mathrm dm$ term. Here, we can see that $M$ will reduce only if $\mathrm dm$ has a positive value. In other words, when time goes forwards, the mass that got thrown out ($m$) is increasing, thus $\mathrm dm$ is positive. In contrast, $M$ ...

5

Sorry, a solid angle is something different than an ordinary angle, see http://en.wikipedia.org/wiki/Solid_angle so it is not measured "with respect to anything". Solid angle $\Omega$ measures the size of a set of directions in the 3-dimensional space via the formula $$\Omega = \frac{A}{R^2}$$ where $A$ is the area of the intersection of all these ...

5

It's an integral over a closed contour (which is topologically a circle). An example from Wikipedia: \begin{align} \oint_C {1 \over z}\,dz & {} = \int_0^{2\pi} {1 \over e^{it}} \, ie^{it}\,dt = i\int_0^{2\pi} 1 \,dt \\ & {} = \Big[t\Big]_0^{2\pi} i=(2\pi-0)i = 2\pi i, \end{align} .

5

The author uses this weird notation $[c:\gamma]$ to represent complex numbers. It means: c is short for the mag­ni­tude $|c|$ of c, $\gamma$ is the phase of c. I have never seen this before either ;-). The author explains it earlier in his book, check out this link.

5

An easy way to see that they are distinct is to consider what happens upon raising (or lowering) all indices. For example, upon lowering, $$T_{ab}{}^{cde}$$ becomes $T_{abcde}$, whereas $$T_{a}{}^{cd}{}_{b}{}^{e}$$ becomes $T_{acdbe}$, and similarly $$T_{a}{}^{cde}{}_{b}$$ becomes $$T_{acdeb}.$$ You need to "slant" the indices so as to keep track ...

5

It is the second, $\psi(x) = \langle x|\psi\rangle$ which is correct. The first, if $x$ is the position operator, is just the position operator acting on the state $|\psi\rangle$. The abstract state $|\psi\rangle$ can be expanded in any basis, using a completion relation: $$|\psi\rangle = \underbrace{\sum_i |i\rangle \langle i|}_{1~=~identity}\psi\rangle$$ ...

5

It is just a matter of notations. For some reasons, physicists tends to note position using a function notation $\hat \psi(x)$, and momentum with a subscript $\hat a_k$. It is just a matter of taste $\hat \psi(x)=\hat \psi_x$. You seem to be confused by the use of a continuous value for the "index" $x$. If you prefer (and I think that is what mathematicians ...

5

There is no significance in the choice between upper- and lower-case $\psi$ (or $\Psi$) to denote a system's wavefunction. The two are used interchangeably and it is the author's discretion to use either symbol. (On the other hand, of course, one shouldn't use the two symbols interchangeably within the same text; if both are used they would refer to ...

5

It's purely notation. Given a real-valued function $f(\mathbf r) = f(x^1, \dots, x^n)$ of $n$ real variables, one defines the derivative with respect to $\mathbf r$ as follows: \begin{align} \frac{\partial f}{\partial \mathbf r}(\mathbf r) = \left(\frac{\partial f}{\partial x^1}(\mathbf r), \dots, \frac{\partial f}{\partial x^n}(\mathbf r)\right) ...

4

I'm not 100% sure about this, but I believe this notation stems from the practice of treating $\psi$ as a wavefunction. In a typical introductory quantum mechanics class, the theory is expressed in terms of continuous functions $\psi(x)$ in the space $\mathbb{R}\to\mathbb{C}$, and operators that act on these functions to produce other functions. So ...

4

Originally, the names were descriptions of series of spectral lines recorded on photographic film. Introduction to Modern Optics By Grant R. Fowles

4

$\partial_t\equiv\frac\partial{\partial t}$ and $\partial^\mu\equiv g^{\mu\nu}\frac\partial{\partial x^\nu}=\left(\sum_{\nu=0}^3g^{\mu\nu}\frac\partial{\partial x^\nu}\right)_{\mu=0}^3$ are differential operators. $\partial^\mu$ is formally contravariant (upper index) and obeys the corresponding transformation laws. $\partial_t$ has a lower index and is (up ...

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