Welcome!

Mathematically rigorous physics, please. A group of physics and mathematics students and professors exploring common ground in mathematical physics.

Faculty advisors:
Physics: Professor André de Gouvêa (ude.nretsewhtron|aevuoged#ude.nretsewhtron|aevuoged) and Professor Tim Tait (ude.nretsewhtron|tiat#ude.nretsewhtron|tiat)
Math: Professor Kevin Costello (ude.nretsewhtron.htam|olletsoc#ude.nretsewhtron.htam|olletsoc), Professor Ezra Getzler (ude.nretsewhtron|relzteg#ude.nretsewhtron|relzteg), and Professor Eric Zaslow (ude.nretsewhtron.htam|wolsaz#ude.nretsewhtron.htam|wolsaz)

Time/Place: Wednesdays, 10:30am-12pm, Tech M166 (note room change!)

Upcoming Lectures:
5/13: Quantum Chern-Simons theory (Yuan Shen)
5/20-5/27: NO SEMINAR because of a conference on topological field theory hosted by the Math Department. However, please check out the workshop talks, especially the introductory ones on Monday 5/18 and Tuesday 5/19

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Some details on Dirac notation
Dirac notation represents elements of a complex vector space $\mathbf{V}$, equipped with a nondegenerate sesquilinear inner product, with objects called "kets": $| \mbox{ket} \rangle , | \mbox{Bob} \rangle, | x + a \rangle$. The text inside the ket ("ket", "Bob" or "x+a") is just a label. Usually, kets are labeled by eigenvalues of a particular operator. For example, if the eigenvalues of the operator $\hat{x}$ are called $x$, then the action of $\hat{x}$ on one of its eigenvectors is written

(1)

\begin{align} \hat{x} | x \rangle = x | x \rangle \end{align}

This is certainly not the high point of notational elegance. However, you will see it all the time in physics literature. Scalars can be put anywhere: if $\alpha \in \mathbb{C}$, then $\alpha | a \rangle$ is the same as $| a \rangle \alpha$.

Elements of the dual space are written as "bras": $\langle a |$. The label on the bra is chosen so that the isomorphism between the vector space and its dual is induced by the inner product:

(2)

\begin{align} \langle a | b \rangle := | b \rangle \cdot | a \rangle \end{align}

So a "bra" and "ket" pair as a "bra-ket" (= bracket). At this point the inner product on $\mathbf{V}$ essentially disappears, and we deal only with dual parings. A physicist might say, "Take an equation involving kets, and dot both sides with $|x \rangle$." This should be parsed as "take the dual pairing with $\langle x |$."

Sesquilinearity requires $\langle a | b \rangle = \langle b | a \rangle ^*$ as usual, but in physics the conjugation happens in the first component:

(3)

\begin{align} | c \rangle = \alpha | a \rangle + \beta | b \rangle \implies \langle c | d \rangle = \alpha^* \langle a | d \rangle + \beta^* \langle b | d \rangle \end{align}

Note that any expression of the form $\langle \ | \ \rangle$ is a (complex) number.

Linear operators act on kets on the left and bras on the right. The action of a linear operator $\hat{O}$ on a bra is defined so that the bracket notation is consistent:

(4)

\begin{align} (\langle b | \hat{O})\, | a \rangle := \langle b |\, (\hat{O} | a \rangle) = \langle b | \hat{O} | a \rangle \end{align}

The adjoint is defined as usual:

(5)

\begin{align} \langle a | \hat{O}^\dagger | b \rangle := (\langle b | \hat{O} | a \rangle)^* \end{align}

The usefulness of Dirac notation is most apparent in equations (4) and (5): we can read an expression like $\langle b | \hat{O} | a \rangle$ as either the pairing of $\hat{O} | a \rangle$ with the dual of $| b \rangle$, or the pairing of $| a \rangle$ with the dual of $\hat{O} | b \rangle$. Even more useful is that various tensor products "appear" natural in this notation. For example, an element of $\mathbf{V} \otimes \mathbf{V}^*$ is written $|a \rangle \langle b |$; this looks like an operator, and indeed it is, since pairing with a ket $|x \rangle$ on the right gives a ket $|a \rangle \langle b | x \rangle$, and pairing with a bra $\langle y |$ on the left gives a bra $\langle y |a \rangle \langle b |$. Similarly, an element of $\mathbf{V} \otimes \mathbf{V}$ is written $|a \rangle \, | b \rangle$; the symbol $\otimes$ is unnecessary since there is no temptation to "multiply" one ket by another.

If the n eigenkets $|a_i \rangle$ of an operator $\hat{O}$ span $\mathbf{V}$, then we have the operator identity

(6)

\begin{align} \hat{I} = \sum_{i=1}^{n}|a_i \rangle \langle a_i | \end{align}

If the spectrum of $\hat{O}$ is continuous, then we have the analogous relation

(7)

\begin{align} \hat{I} = \int|a_x \rangle \langle a_x |\,dx \end{align}

Inserting this relation (as the identity operator) somewhere in an equation can make certain results seem almost trivial in this notation. For example, suppose we wanted to express a ket $| \Psi \rangle$ in terms of eigenkets of the position operator (ignoring, for the moment, that these kets don't really belong to the Hilbert space $L^2(\mathbb{R})$). We then have

(8)

\begin{align} | \Psi \rangle = \hat{I} | \Psi \rangle = \int|x \rangle \langle x | \Psi \rangle \,dx, \end{align}

and we identify the numbers $\langle x | \Psi \rangle$ with the function $\Psi(x)$. We can also express the pairing $\langle \Phi | \Psi \rangle$ between two vectors as

(9)

\begin{align} \langle \Phi | \Psi \rangle = \int \langle \Phi |x \rangle \langle x | \Psi \rangle \,dx = \int \Phi^*(x) \Psi(x) \, dx \end{align}