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35

Coming from a math perspective, I would define a dimension as "any property which is orthogonal to all other properties." "Orthogonal" here means you cannot get to one property by applying scalar operations on another. For example, the x-axis dimension can never become a y-axis value, and similarly for time vs. spatial dimensions. For that matter, ...


33

A second-order tensor can be represented by a matrix, just as a first-order tensor can be represented by an array. But there is more to the tensor than just its arrangement of components; we also need to include how the array transforms upon a change of basis. So tensor is an n-dimensional array satisfying a particular transformation law. So, yes, a ...


27

In this context, I usually explain it (non-mathematically) by saying that the number of dimensions is the number of values you need to specify where an event occurs. For most people this involves space and time (but for particle physicists it might involve more values ;). Anyway, certainly even people before Einstein would need to specify the time as well ...


21

Matrices are often first introduced to students to represent linear transformations taking vectors from $\mathbb{R}^n$ and mapping them to vectors in $\mathbb{R}^m$. A given linear transformation may be represented by infinitely many different matrices depending on the basis vectors chosen for $\mathbb{R}^n$ and $\mathbb{R}^m$, and a well-defined ...


18

The slave particle approach is based on the assumption of spin-charge separation in the strongly correlated electron systems (typically Mott insulators). It was proposed that the electrons can decay into spinons and chargons (holons/doublons). But to preserve the fermion statistics of the electrons, the spinon-chargon bound state must be fermionic, so the ...


16

Quantum mechanics "lives" in a Hilbert space, and Hilbert space is "just" an infinite-dimensional vector space, so that the vectors are actually functions. Then the mathematics of quantum mechanics is pretty much "just" linear operators in the Hilbert space. Quantum mechanics Linear algebra ----------------- -------------- wave function vector ...


15

The dual of a tensor you refer to is the Hodge dual, and has nothing to do with the dual of a vector. The word "dual" is used in too many different contexts, and in this case it is even used the same $*$ symbol. One usually specifies "Hodge dual", or "Hodge star operator", to avoid confusion. Both these "duals" are isomorphisms between vector spaces endowed ...


13

First, just to be sure about the answers to this particular problem: the eigenvalues of the $4\times 4$ matrix are $$0,\quad U\quad {\rm and}\quad U/2\pm \sqrt{(U/2)^2+4t^2}$$ When expanded to the first nontrivial order, the last two eigenvalues are $$ 0 - \frac{4t^2}U \quad {\rm and} \quad U+\frac{4t^2}U. $$ Note that the corrections to the energy arise ...


13

The wording used in your textbook was sloppy. $A$ acts as $A^*$ on a bra, as $\langle u\rvert A\lvert v\rangle:=\langle u\lvert Av\rangle~$ is the same as $\langle u\rvert A\lvert v\rangle=\langle A^*u\lvert v\rangle~$, by definition of the adjoint. The latter formula also shows that $\langle A^*u\rvert=\langle u\rvert A$. Everything becomes very simple ...


13

There are at least three notions of basis depending on the mathematical structure you are considering. I will quickly discuss three cases relevant in physics (topological vector spaces are relevant too, but I will not consider them for the shake of brevity). (1) Pure algebraic structure (i.e. vector space structure over the field $\mathbb K=$ $\mathbb R$ ...


12

Your doubt is not ridiculous, it is probably simply due to the confused way often mathematics is taught in physics. (I am a physicist too and, during my career, I had to bear ridiculous misconceptions, wasting lot of time in tackling non-existent pseudo-mathematical problems instead of focusing on genuine physical issues). There are sensible mathematical ...


12

If the Hilbert space of the system in question is finite-dimensional, then in a given basis for the Hilbert space, the Hamiltonian (and every other observable for that matter), will be represented by a matrix. If the Hilbert space is infinite-dimensional, the situation is a bit different. In Quantum Mechanics, we typically assume that the Hilbert spaces we ...


12

Call $u_1, u_2, u_3, u_4$ the eigenvectors described by you, respectively. Your claims are all right, but realize that both $u_1$ and $u_2$ share the same eigenvalue, that is $1$, i.e., $Pu_1=u_1$ and $Pu_2=u_2$. Hence, any linear combination of $u_1$ and $u_2$ will also be eigenvectors with the same eigenvalue $1$. Try to find eigenvectors of $H$ of the ...


12

Does this mean that the operator $\hat O$ (an observable) is special in some way? I believe it means there is no such $\hat O$. If $\hat O$ corresponds to an observable, we require the eigenvalues to be real. Let $|o\rangle$ be an eigenket of $\hat O$ with real eigenvalue $o$: $$\hat O |o\rangle = o |o\rangle$$ Now consider the following $$\hat O ...


11

There are two concepts of duality for vector spaces. One is the algebraic dual that is the set of all linear maps. Precisely, given a vector space $V$ over a field $\mathbb{K}$, the algebraic dual $V_{alg}^*$ is the set of all linear functions $\phi:V\to \mathbb{K}$. This is a subset of $\mathbb{K}^V$, the set of all functions from $V$ to $\mathbb{K}$. The ...


11

The ground state of the harmonic oscillator $|0\rangle$ obeys $$a|0\rangle = 0$$ which means that the action of $a$ can't be undone: once you act with it on a state, you set to zero the coefficient in front of $|0\rangle$ in the decomposition into occupation eigenstates. Any candidate inverse operator $a^{-1}$ acting on zero will give you zero again; you ...


10

Hint: When an eigenvalue for an operator $P$ is degenerate, there are more than one way to chose a set of eigenvectors. If the other commuting operator $H$ lifts that degeneracy, there will be a preferred choice of common eigenvectors. More generally, a set of diagonalizable operators commutes if and only if the set is simultaneously diagonalizable.$^1$ ...


10

Observe that \begin{equation} \sigma_{z} = \begin{pmatrix}1&0\\0&-1\end{pmatrix} = \exp(B) = \sum_{r=0}^{\infty} \frac{B^{r}}{r!} \end{equation} with \begin{equation} B = i\pi\begin{pmatrix}2m&0\\0&2n+1\end{pmatrix}, \end{equation} where $m,n\in\mathbb{Z}$. Next, notice that \begin{equation} \sigma_{x} = U \sigma_{z} U^{\dagger} ...


9

Flip back a page; Dirac uses real to mean Hermitian ($A^{\dagger} = A$) when talking about linear operators. So you can see that even if $A$ and $B$ are Hermitian, $AB$ won't be Hermitian unless they commute, whereas those linear combinations will be.


8

It is easy. Assume that $\psi \in \cal H$ is normalized to $1$. In this case, $|\psi\rangle\langle\psi|$ it is nothing but the orthogonal projector $P_\psi$ onto the one-dimensional linear space generated by the vector $\psi$. Putting $|\psi\rangle$ and $\langle\psi|$ together simply means to exploit the tensor product. If $\phi' \in \cal H'$ the ...


8

Let $\{|i\rangle\}$ be an orthonormal basis for the Hilbert space of the system. Then the trace of an operator $O$ is given by (See the Addendum below) \begin{align} \mathrm {tr}(O) = \sum_i \langle i|O|i\rangle \end{align} For a given state $|\psi\rangle$, we define an operator $P_\psi$ by \begin{align} P_\psi|\phi\rangle = ...


8

There is quite a lot of very important information hidden in the term hermitian. For an operator $A$ on a finite-dimensional Hilbert space $\mathcal H$, one can show that there exists an orthonormal basis for the Hilbert space consisting of eigenvectors of the operator $A$. Moreover, one can show that the eigenvalues corresponding to these eigenvectors ...


8

The expectation value of energy is something else than the energy in a particular experiment. With your choice of the initial states, the photons emitted (negative difference) or absorbed (positive difference) will have energies either $$ E_1-E_0 \text{ or } E_2-E_0 \text{ or } E_1-E_5 \text{ or } E_2-E_5 $$ If each of the four transitions were equally ...


7

For several years I have been teaching Clifford (geometric) algebra as part of the Vector Analysis Course for undergraduate physics majors in Ateneo de Manila University. I strictly use Cl_{n,0}, even for Special Relativity. 18-year old students do not complain how difficult geometric algebra is. They just learn the math and the geometric interpretations: ...


7

Ron Maimon is entirely correct when he says that GA is precisely Clifford algebra, as any book or paper using the phrase "Geometric Algebra" is sure to say. But I think he misses both the point of the question and the point of "GA". The question I'll paraphrase the question as: Is GA a good, pedagogical way to introduce the mathematical side of physics to ...


7

Well, learn linear algebra. An advanced text (on linear algebra over "field" number systems) is these lecture notes [pdf] from UC Davis. Once you get that done, you should study differential equations. Or if you want to skip ahead, perhaps Fourier analysis. A free reference would be my notes [pdf]. It's mildly physics-oriented, but connects the ideas back ...


7

Consider first the matrix representation of a single Grassmann number as $\theta = \left[\begin{matrix} a & b \\ c & d \end{matrix}\right]$ where $\theta^2 = \left[\begin{matrix} a^2+bc & b(a+d) \\ c(a+d) & d^2+bc \end{matrix}\right] = 0 \Leftrightarrow \theta = \left[\begin{matrix} i\sqrt{bc} & b \\ c & -i\sqrt{bc} ...


6

Length and distance are not vector quantities (they are scalar quantities), but position and displacement are vector quantities (at least according to common terminological conventions). Here is how all of these are defined. Note that I am restricting the discussion here to vectors in three-dimensional Euclidean space $\mathbb R^3$. Every point in ...


6

I suggest to think of it like this: $$\left|\psi\right.\rangle = \left(\begin{array}{c}\psi_1\\\psi_2\\\cdots\\\psi_n\end{array}\right)\;,\qquad \langle\left.\psi\right| = \left(\begin{array}{c}\psi_1^*\,\psi_2^*\,\cdots\,\psi_n^*\end{array}\right) $$ And using standard rule ("every element in row per every element in column") for matrix multiplication. ...



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