Amplitudes at SCGP

The Simons Center for Geometry and Physics in Stony Brook is going to be running the program Physics and Mathematics of Scattering Amplitudes during the Fall 2013 semester. It started last week with talks by Henrietta Elvang and Johannes Henn. Sadly, I am going to miss the talks in person. But thankfully, there are going to be videos of most of the talks. The talks from last week can be found here and here.

Defense Success!

Yesterday, I successfully defended my dissertation in Stony Brook. It was a great moment for me. I thank my committee for helping me understand my own work. You can read my dissertation here, and my slides here. Maybe I will talk more about my work soon.

Three-body interaction for particle action

Consider a scalar quantum field theory with the following interaction vertices:\[g_{F} |\Phi|^{2} A + \frac{h_{F}}{6}A^{3}\]where \(g_{F}\) and \(h_{F}\) are two coupling strengths for cubic interactions. In the particle theory we can consider a three-body system with a three-body interaction (i.e. not pair-wise) that has the form\[S_{3}\left[ q_{14}, \, q_{25}, \, q_{36} \right] \equiv g_{P}^{3} h_{F} \int\limits_{0}^{T_{14}} \mathrm{d}\tau \int\limits_{0}^{T_{25}} \mathrm{d}\sigma \int\limits_{0}^{T_{36}} \mathrm{d}\rho \int \mathrm{d}Y \, G_{A}\left[ q_{14}(\tau)| Y \right] G_{A}\left[ q_{25}(\sigma)| Y \right] G_{A}\left[ q_{36}(\rho)| Y \right] \]In the eikonal approximation we will get a three-body Gram invariant from this term in the action.

Gram invariants in momentum space

Given two vectors \(k_{1}\) and \(k_{2}\) in a $D$-dimensional euclidean space we can consider the parallelogram formed with them. The area of this parallelogram is given by the square root of the determinant of the Gram matrix \(G_{12}\):\[G_{12} = \begin{pmatrix} k_{1}^{2} & k_{1} \cdot k_{2} \\ k_{1} \cdot k_{2} & k_{2}^{2} \end{pmatrix} \quad \Longrightarrow \quad \det{(G_{12})} = k_{1}^{2} k_{2}^{2} - (k_{1} \cdot k_{2})^{2}\]We will denote the magnitude of each vector by \(|k_{i}| \equiv m_{i}\). Then it follows that\[\det{(G_{12})} = \left[m_{1}m_{2} - k_{1} \cdot k_{2} \right]\left[m_{1}m_{2} + k_{1} \cdot k_{2} \right]\]Introducing the invariant \(s_{12} \equiv (k_{1} + k_{2})^{2}\) we can write\[\det{(G_{12})} = \frac{1}{4} \left[ s_{12} - (m_{1} - m_{2})^{2} \right] \left[ (m_{1} + m_{2})^{2} - s_{12} \right]\]Hence, the area of the parallelogram generated by \(k_{1}\) and \(k_{2}\) is\[A_{12} = \frac{1}{2} \sqrt{\left[ s_{12} - (m_{1} - m_{2})^{2} \right] \left[ (m_{1} + m_{2})^{2} - s_{12} \right]}\]Note that this area is real in the domain\[(m_{1} - m_{2})^{2} \leq s_{12} \leq (m_{1} + m_{2})^{2}\]The notation that we have used is very suggestive. If the two vectors were momentum vectors, then \(m_{i}\) would correspond to masses and \(s_{12}\) would correspond to a Mandelstam invariant. Then \( (m_{1}+m_{2})^{2} \) is a mass threshold and \( (m_{1} - m_{2})^{2} \) a mass pseudothreshold.

We can study the case with three vectors. Now we have three masses and hence three three-body mass pseudothresholds:\[(m_{1} - m_{2} + m_{3})^{2} \qquad (m_{1} + m_{2} - m_{3})^{2} \qquad (m_{1} - m_{2} - m_{3})^{2}\]We also have three (two-body) Mandelstam invariants:\[s_{12} \equiv (k_{1} + k_{2})^{2} \qquad s_{23} \equiv (k_{2} + k_{3})^{2} \qquad s_{31} \equiv (k_{3} + k_{1})^{2}\]It might be convenient to introduce a three-body Mandelstam invariant:\[t_{123} \equiv (k_{1} + k_{2} + k_{3})^{2} = s_{12} + s_{23} + s_{31} - m_{1}^{2} - m_{2}^{2} - m_{3}^{2}\]Note that\[4 (m_{1}^{2} + m_{2}^{2} + m_{3}^{2}) = (m_{1} + m_{2} + m_{3})^{2} + (m_{1} - m_{2} + m_{3})^{2} +(m_{1} + m_{2} - m_{3})^{2} + (m_{1} - m_{2} - m_{3})^{2}\]Now the three vectors will form a parallelepiped whose volume is given by the square root of the Gram determinant:\[V_{123} \equiv \sqrt{\det{(G_{123})}}\]What is driving me nuts is how to write this determinant in a nice form that involves the above invariants. Naively we have\[\det{(G_{123})} = m_{1}^{2}m_{2}^{2}m_{3}^{2} - m_{1}^{2} (k_{2} \cdot k_{3})^{2} - m_{2}^{2}(k_{1} \cdot k_{3})^{2} - m_{3}^{2} (k_{1} \cdot k_{2})^{2} + 2 (k_{1} \cdot k_{2})(k_{2} \cdot k_{3})(k_{3} \cdot k_{1})\]The secret might be with traces.

Multivariate weight functions

These two look promising. One for bivariate Hermite polynomials:\[ w_{H}(x, \, y|z) = \frac{1}{\sqrt{1 - z^{2}}} \exp{\left( \frac{2 x y z -x^{2} - y^{2}}{1 - z^{2}} \right)} \qquad -1 < z < 1\]and one for bivariate Gegenbauer polynomials:\[w_{G}(x, \, y|z) = \frac{1}{\sqrt{1 - z^{2}}} \left( \frac{1 + 2xyz - x^{2} - y^{2} - z^{2}}{1 - z^{2}} \right)^{\alpha} \qquad -1 < z < 1\]

Four-point Kinematics: Momentum Basis

We have four external momentum vectors:\[k_{1} \qquad k_{2} \qquad k_{3} \qquad k_{4}\]In the $s$-channel we take $k_{1}$ and $k_{2}$ to be incoming. The external momenta are constrained by the conservation condition:\[k_{1} + k_{2} = k_{3} + k_{4}\]and the on-shell conditions:\[k_{1}^{2} = -m_{1}^{2} \qquad k_{2}^{2} = -m_{2}^{2} \qquad k_{3}^{2} = -m_{3}^{2} \qquad k_{4}^{2} = -m_{4}^{2}\]The Mandelstam invariants are defined as\[s \equiv - (k_{1} + k_{2})^{2} \qquad t \equiv - (k_{1} - k_{4})^{2} \qquad u \equiv - (k_{1} - k_{3})^{2}\]Note that these three invariants are not independent,\[s + t + u = m_{1}^{2} + m_{2}^{2} + m_{3}^{2} + m_{4}^{2}\]Using momentum conservation, we can also write\[s = -(k_{3} + k_{4})^{2} \qquad t = -(k_{3} - k_{2})^{2} \qquad u = -(k_{4} - k_{2})^{2}\]We now write all other inner products in terms of the massess and the three Mandelstam invariants:\[k_{1} \cdot k_{2} = \frac{1}{2} \left[ m_{1}^{2} + m_{2}^{2} - s \right] \qquad k_{1} \cdot k_{3} = \frac{1}{2} \left[ u - m_{1}^{2} - m_{3}^{2} \right] \qquad k_{1} \cdot k_{4} = \frac{1}{2} \left[ t - m_{1}^{2} - m_{4}^{2} \right]\]\[k_{2} \cdot k_{3} = \frac{1}{2} \left[ t - m_{2}^{2} - m_{3}^{2} \right] \qquad k_{2} \cdot k_{4} = \frac{1}{2} \left[ u - m_{2}^{2} - m_{4}^{2} \right]  \qquad k_{3} \cdot k_{4} = \frac{1}{2} \left[m_{3}^{2} + m_{4}^{2} - s \right]\]

Massive Scalar Propagator in Euclidean Space

In Euclidean space, the propagator for a massive scalar field with mass \(M > 0\) is\[G(x|y) \equiv \left( \frac{\hbar}{\mu} \right) \int\limits_{0}^{\infty} \mathrm{d}\xi \left( \frac{\mu}{\hbar \xi} \right)^{D/2} \exp{\left( -\frac{\mu (x - y)^{2}}{2 \hbar \xi} - \frac{M^{2} \xi}{2 \hbar \mu} \right)} \]where $D$ is the dimension of space and $\mu$ is a constant with units of mass. We have used a Schwinger parameter $\xi$ to write the propagator. This Schwinger parameter has units of length. The integral over $\xi$ can be performed explicitly to give\[G(x|y) = 2 \left( \frac{\mu^{2}}{\hbar^{2}} \right)^{\Delta} \left( \frac{\hbar}{\mu |x - y|} \right)^{\Delta} \left( \frac{M}{\mu} \right)^{\Delta} K_{\Delta} \left( \frac{M |x - y|}{\hbar} \right) \qquad \Delta \equiv \frac{D - 2}{2}\]where $K_{\Delta}$ is a modified Bessel function.

The semiclassical propagator is defined by taking $\hbar \rightarrow 0$. This corresponds to either using the asymptotic expansion for the Bessel function, or performing the integral over $\xi$ with saddle-point approximations. The result is\[G(x|y) \approx \left( \frac{\mu^{2}}{\hbar^{2}} \right)^{\Delta} \left( \frac{M}{\mu} \right)^{(D - 3) / 2} \left( \frac{\hbar^{2}}{\mu^{2} (x - y)^{2}} \right)^{(D - 1)/2} \exp{\left(- \frac{M |x - y|}{\hbar} \right)}\]We write this as an infinite series:\[G(x|y) \approx \left( \frac{\mu^{2}}{\hbar^{2}} \right)^{\Delta} \sum_{n = 0}^{\infty} \frac{(-1)^{n}}{\Gamma(n + 1)} \left( \frac{M}{\mu} \right)^{(2\Delta + 2n - 1)/2} \left( \frac{\hbar^{2}}{\mu^{2} (x - y)^{2}} \right)^{(2 \Delta - n + 1)/2}\]

Numerology with scalar fields

Today is a good day to do some numerology, being March 14th and Einstein's birthday. Consider a quantum field theory in \(D\) spacetime dimensions with scalars \(\varphi\) and an interaction of the form \(g_{n}\varphi^{n}\). We are going to measure everything in units of mass and units of \( \hbar \). Taking the speed of light to be dimensionless leads to length, time, momentum and energy written in terms of mass and \( \hbar \). For example, the action has units of \( \hbar \). The scalar field has units\[ \left[ \varphi \right] =  \left( \frac{3 - D}{2} \right) \left[ \hbar \right] + \left( \frac{D - 2}{2} \right) \left[ M \right] \]The dimension of the coupling satisfies\[ \left[ g_{n} \right] = \left[ \hbar \right] - n \left[ \varphi \right] - D \left[ L \right] \]Using the dimension for \(\varphi\) and the fact that \(\left[ L \right] = \left[ \hbar \right] - \left[ M \right] \) we get\[\left[ g_{n} \right] = \left( \frac{(n - 2)D + 2 - 3n}{2} \right) \left[ \hbar \right] + \left( \frac{(2-n)D + 2n}{2} \right) \left[ M \right]\]Here comes the numerology. For \(n = 3\) one gets\[\left[ g_{3} \right] = \left( \frac{D - 7}{2} \right) \left[ \hbar \right] + \left( \frac{6 - D}{2} \right) \left[ M \right]\]For \(n = 4\) one gets\[\left[ g_{4} \right] = \left( D - 5 \right) \left[ \hbar \right] + \left( 4 - D \right) \left[ M \right]\]And finally, for \(n = 6\) one get\[\left[ g_{6} \right] = 2\left( D - 4 \right) \left[ \hbar \right] + 2\left( 3 - D \right) \left[ M \right]\]In all three cases the mass dimension of the coupling vanishes in a dimension lower than the \(\hbar\) dimension. For \(n = 3\) we have vanishing mass dimension in \(D = 6\) and vanishing \(\hbar\) dimension in \(D = 7\). For \(n = 4\) we have vanishing mass dimension in \(D = 4\) and vanishing \(\hbar\) dimension in \(D = 5\). And finally for \(n = 6\) we have vanishing mass dimension in \(D = 3\) and vanishing \(\hbar\) dimension in \(D = 4\). These three cases coincide with the three canonical cases of \(adS_{D+1} / CFT_{D}\). Indeed, \(\mathcal{N} = 4\) super Yang-Mills in four spacetime dimensions contains scalar fields that interact with a quartic vertex and ABJM theory in three spacetime dimensions has scalar fields that interact via a sextic vertex. I guess this means that the theory related to the \(\mathcal{N} = (2, \, 0)\) theory in six spacetime dimensions has scalars interacting via a cubic vertex.

Actually, the coefficient of \(\left[M\right]\) in \(D\) dimensions equals the negative of the coefficient of \(\left[ \hbar \right]\) in \(D+1\) dimensions for any value of \(n\). However, only for the three cases above we have integer solutions.

If one performs the same analysis on the coupling constant in Yang-Mills theory, the dimension comes out like\[\left[ g_{YM} \right] = \left( \frac{D - 3}{2} \right)\left[\hbar \right] + \left( \frac{4 - D}{2} \right) \left[ M \right]\]However, the coupling appears in the field theory action in the combination \(g_{YM} / \hbar\) and this leads to\[\left[ g_{YM} \right] - \left[\hbar \right] = \left( \frac{D - 5}{2} \right)\left[\hbar \right] + \left( \frac{4 - D}{2} \right) \left[ M \right]\]

Coulomb electrostatics in arbitrary spatial dimensions

In three spatial dimensions, the electrostatic potential \(\phi(\mathbf{x})\) satisfies Poisson's equation:\[\left(\frac{\partial}{\partial \mathbf{x}} \cdot \frac{\partial}{\partial \mathbf{x}} \right) \phi(\mathbf{x}) = - \frac{\rho(\mathbf{x})}{\varepsilon_{0}}\]The solution to this partial differential equation can be written down in terms of the Green function for Laplace's operator:\[\phi(\mathbf{x}) = \phi_{0}(\mathbf{x}) + \int dy \left[ G(\mathbf{x}, \, \mathbf{y}) \rho(\mathbf{y})\right] \]where \(\phi_{0}\) is a harmonic function and \(G\) satisfies\[\left(\frac{\partial}{\partial \mathbf{x}} \cdot \frac{\partial}{\partial \mathbf{x}} \right) G(\mathbf{x}, \, \mathbf{y}) = -\frac{1}{\varepsilon_{0}}\delta(\mathbf{x} - \mathbf{y})\]In order to find the Green function, we write it as the Fourier transform of another function:\[G(\mathbf{x}, \, \mathbf{y}) = \int \int dbdc \left[ g(\mathbf{b}, \, \mathbf{c}) \exp{\left( -i \mathbf{x} \cdot \mathbf{b} +i\mathbf{y} \cdot \mathbf{c}\right)} \right]\]This leads to\[g(\mathbf{b}, \, \mathbf{c}) = \frac{1}{\varepsilon_{0}} \frac{\delta(\mathbf{b} - \mathbf{c})}{\mathbf{b}^{2}}= \frac{1}{2\varepsilon_{0}} \delta(\mathbf{b} - \mathbf{c}) \int_{0}^{\infty}d\tau \, \exp{\left(- \frac{\tau}{2}\mathbf{b}^{2} \right)} \]Then, the Green function yields\[G(\mathbf{x}, \, \mathbf{y}) = \frac{1}{2\varepsilon_{0}} \int_{0}^{\infty} d\tau \left( \frac{1}{\sqrt{\tau}} \right)^{3} \exp{\left(-\frac{1}{2\tau}(\mathbf{x} - \mathbf{y})^{2} \right)}\]This expression can be generalize to \(D\) spatial dimensions: \[G(\mathbf{x}, \, \mathbf{y}) = \frac{1}{2 \varepsilon_{D}}\int_{0}^{\infty} d\tau \left( \frac{1}{\sqrt{\tau}} \right)^{D} \exp{\left(- \frac{1}{2\tau} (\mathbf{x} - \mathbf{y})^{2} \right)}\]where we have introduced \(\varepsilon_{D}\) as the \(D\)-dimensional analog of \(\varepsilon_{0}\). The Coulomb potential corresponds to a localized charge density:\[\rho(\mathbf{x}) = e \delta(\mathbf{x} - \mathbf{x}_{e})\]In \(D\) spatial dimensions, the Coulomb potential is\[\phi_{C}(\mathbf{x}) = \frac{e}{2 \varepsilon_{D}}\int_{0}^{\infty} d\tau \left( \frac{1}{\sqrt{\tau}} \right)^{D} \exp{\left(- \frac{1}{2\tau} (\mathbf{x} - \mathbf{x}_{e})^{2} \right)} = \frac{e}{2 \varepsilon_{D}} \left( \frac{2}{(\mathbf{x} - \mathbf{x}_{e})^{2}} \right)^{\nu}\Gamma(\nu) \qquad \nu \equiv \frac{D - 2}{2}\]This expression reduces to the familiar form when \(D = 3\).

In three spatial dimensions, the Green function can be written as a Legendre series. For arbitrary spatial dimension \(D\) the Legendre series generalizes to a Gegenbauer series. With \(|\mathbf{x}| > |\mathbf{y}|\) this yields\[G(\mathbf{x}, \, \mathbf{y}) = \frac{\Gamma(\nu)}{2\varepsilon_{D}}\left(\frac{2}{|\mathbf{x}| |\mathbf{y}|}\right)^{\nu}\sum_{n = 0}^{\infty}\eta^{n+\nu}C_{n}^{\hphantom{n}\nu}(\xi)\]where\[\eta \equiv \frac{|\mathbf{y}|}{|\mathbf{x}|} \qquad \xi \equiv \frac{\mathbf{x} \cdot \mathbf{y}}{|\mathbf{x}| |\mathbf{y}|}\]

Multivariate polynomials

While doing research, I came across some interesting multivariate versions of familiar orthogonal polynomials like Legendre or Chebyshev. In the end, they appear to not be very useful for my particular research problem. Still, I want to write a bit about what I found because they are kind of cute.

Alday and Maldacena I

Luis Alday and Juan Maldacena opened the floodgates in 2007 with their work on scattering amplitudes at strong coupling. By finding the solution to the equation of motion for bosonic strings that move in $adS_{5}$, Alday and Maldacena were able to make contact with something that looked like the BDS ansatz for scattering amplitudes of gluons in $\mathcal{N} = 4$ super Yang-Mills. The connection between tree-level string theory in $adS_{5}$ and strongly-coupled super Yang-Mills is possible thanks to the anti de-Sitter / conformal field theory correspondence.

The work of Alday and Maldacena is important for many reasons. First, it makes contact with gauge theory scattering amplitudes at strong coupling, something that is naively inaccessible with perturbation theory. Second, the way the Alday-Maldacena amplitude was obtained uncovered a link between scattering amplitudes and expectation values of certain null Wilson loops. This in turn lead to uncovering the Yangian symmetry of the planar sector.

The bold claim is that the scattering amplitude at strong coupling in the gauge theory side of adS/CFT corresponds to the classical limit of a scattering amplitude in the string theory side.

String theory is usually first-quantized. This means that the action functional involves the geometric variables that describe (in some limit) the classical dynamics of a string and not some fields in spacetime. That is, \[S\left[ X \right] = \int d^{2}\sigma \, \mathcal{L}\left[X, \, \partial X \right] \] When a string moves through spacetime, it traces out a surface called the worldsheet. The interaction of strings can be described by considering a disk with a certain amount of punctures on the boundary of the disk. The number of punctures corresponds to the number of external states. It is at this punctures that vertex operators are inserted carrying the information of the external states. The take-away from this is that, classically, the boundary conditions of the string worldsheet contain the information about the external states of the process. The string action will lead to equations of motions. To solve this equations of motions one needs to specify boundary conditions. Alday and Maldacena solved the equations of motion for a string with boundary conditions such that the worldsheet described the scattering of four external bosons. Later, this was generalised to any number of external states.

But the strings that Alday and Maldacena studied moved in $adS_{5}$. The equations of motion are hard to solve given the boundary conditions. It proved useful to perform a coordinate transformation to simplify the boundary conditions. The transformation that AM performed had the same form as a T-duality transformation. The original problem had as string worldsheet a surface that was pinched at four points. Under the T-duality, the string worldsheet becomes a surface that ends on a four-sided polygon. The sides of the polygon are null since they are related to the momentum of the external states, which are massless. The problem of finding a surface with this boundary condition is apparently simpler and AM give a classical solution.

With the classical solution $X_{cl}(\sigma)$ at hand, one can obtain the value of the action functional at this configuration. Since the strings are relativistic, the action is proportional to the area of the worldsheet. The answer for AM was divergent. After appropriate regularization an answer for the classical action $S_{cl}$ was given. Since the amplitude is related to a path-integral of the form \[A_{4} = \int DX \left(V_{1} V_{2} V_{3} V_{4} e^{iS\left[X\right]}\right)\] the semiclassical approximation to this amplitude is of the form \[A_{4} \sim \exp{\left( i S_{cl} \right)}\] Alday and Maldacena found that the classical action $S_{cl}$ has the form \[S_{cl} = S_{div}(s) + S_{div}(t) + S_{finite}(s, \, t)\] with all of three terms in the right-hand side being functions of the `t Hooft coupling $\lambda$ too. Both the divergent and the finite part agree exactly with the BDS ansatz.

Hertz potential

A few months ago I learned about the Hertz potential. If one works with the Lorenz gauge, \[\partial^{\mu} A_{\mu} = 0\] then the gauge field can be written in terms of an antisymmetric tensor
\[A_{\mu} = \partial^{\nu} H_{\nu \mu}\] In $d = 4$ we can have electric and magnetic gauge fields. Similarly, one can have electric and magnetic Hertz potentials.

Non-local Bi-linear term

I have been trying to understand the effects from non-locality with the following "generalization" of the harmonic potential: \[S\left[q(t), \, J(t) \right] = \int dt \left[-\frac{1}{2}m\dot{q}^{2} - q \cdot J(t)\right] + \frac{1}{2}m\omega^{2} \int dt \int ds \left[ q(t) \cdot K(t, \, s) \cdot q(s) \right] \] where the kernel is \[K_{jk}(t, \, s) = \delta_{jk}\sqrt{\frac{1}{2 \pi \epsilon}} \exp{\left(-\frac{(t-s)^{2}}{2\epsilon}\right)}\] I am interested in this kernel since it becomes the Dirac delta kernel in the limit $ \epsilon \rightarrow 0 $. The kernel $K(t, \, s)$ also appears in the Weierstrass transform of $q(t)$: \[w(s) = \int dt \left[ K(s, \, t) \cdot q(t) \right]\] Since taking $\epsilon \rightarrow 0$ leads to the Dirac delta kernel, this limit identifies the function $w$ with $q$.

Schroedinger

An abstract way to state the Schrödinger equation is \[ (H - E) \left| \psi \right\rangle = 0 \] with $ H $ the Hamiltonian operator, $ E $ the energy operator and $\left| \psi \right\rangle$ the state vector of the system. Notice that this equation does not say anything about the wavefunction: it only involve things that live abstractly in the Hilbert space. In order to be more concrete, we introduce a complete basis for the Hilbert space. Given a state vector $ \left| \psi \right\rangle $ we conveniently expand it in terms of a basis vectors $ \left| x \right\rangle $ as \[ \left| \psi \right\rangle = \sum_{x} \psi_{x} \left| x \right\rangle \] Sometimes the sum over $ x $ is discrete, sometimes it is continuous. Usually it is some familiar quantity like position or momentum. But not always! The $\psi_{x}$ are the complex coefficients that determine the state: \[ \psi_{x} = \left\langle x | \psi \right\rangle \] We can also find the matrix elements of an operator: \[ O_{xy} = \left\langle x | O | y \right\rangle \] So once we have choosen a basis for the state vectors and the operators, we can consider matrix-like equations of the form \[ \sum_{y} O_{xy} \psi_{y} = 0 \] In component form, the Schrödinger equation is \[ \sum_{y} (H - E)_{xy} \psi_{y} = 0 \] We need to incorporate the notion of "evolution". There are two useful ways to parameterize evolution: in time or in energy. Time and energy are conjugate quantities. This means that we can do either one but not both simultaneously. If we choose to describe evolution in time, then (in the Schrödinger picture) states will carry a time label $t$. A state $ \left| \psi \right\rangle $ with a time label $t_{A}$ is related to the same state with a different time label $t_{Z}$ via a unitary transformation: \[ \left| \psi, \, t_{Z} \right\rangle = U(t_{Z}, \, t_{A}) \left| \psi, \, t_{A} \right\rangle \] Again, this is an abstract equation. In terms of components this reads \[ \psi_{z}(t_{Z}) = \sum_{a} G_{za} (t_{Z}, \, t_{A}) \psi_{a} (t_{A}) \] We have stumble upon a very important object, the fundamental "matrix" $G_{za}(t_{Z}, \, t_{A})$. This object "propagates" the coefficients of a state vector in time. Let me be more concrete: in the position basis, the fundamental amplitude reads \[ G(z, \, a | t_{Z}, \, t_{A}) = \left\langle z | U(t_{Z}, \, t_{A}) | a \right\rangle \] This is the amplitude that Feynman wrote as a sum over trajectories: a path integral.

Let us note that $ G_{za} (t_{Z}, \, t_{A}) $ will satisfy the same Schrödinger equation that $\psi_{z}(t_{Z})$ does. In the Van Vleck approximation, the fundamental amplitude is approximated with the Van Vleck amplitude $ V_{za} (t_{Z}, \, t_{A})$. I will discuss this soon.

Lecture 1: Motion in One Dimension

What is motion? Why does motion happen? What is the description of the motion?

Summer Physics

I am currently teaching a three week course on introductory physics to incoming freshmen. The idea is to give the students a good impression of how a college physics course looks like. Since the student's background is quite varied, I decided to start with basic stuff: motion. During this week we covered motion in one dimension and vectors in two dimensions. Next week we are going to cover motion in two dimensions (probably projectile motion only...) and forces. Hopefully during the third week I will be able to mention something about energy and momentum and conservation laws.

Mandelung and Van Vleck

The Schrödinger equation describes the time evolution of quantum states:

\[ H(t) \left| \psi , \, t \right\rangle = i \hbar \frac{\partial}{\partial t} \left| \psi , \, t \right\rangle \]

Expanding the state in the coordinate basis
\[ \left| \psi, \, t \right\rangle = \int dx \, \psi(x, \, t) \left| x \right\rangle \]
allows us to write the Schrödinger equation for the wavefunction $ \psi(x, \, t) $. Since we are working with vectors in a Hilbert space, the wavefunction is a complex number. One could do three things. First one could write the complex wavefunction in terms of real and imaginary parts:

\[ \psi = \psi_{1} + i \psi_{2} \]

Second, one could write the complex wavefunction in terms of its magnitude and phase:

\[ \psi = \sqrt{\rho} \exp{(i \theta)} \]

And third, one could shut-up and work with the complex wavefunction like everyone else does. It turns out that writing the wavefunction in terms of its magnitude and phase is known as the Mandelung ansatz. This is related to what one does in the Van Vleck approximation where one writes the phase of the wavefunction as

\[ \theta = -\frac{1}{\hbar} \Sigma \]

and looks at the limit $\hbar \rightarrow 0$. The Schrödinger equation becomes the Hamilton-Jacobi equation with a current continuity equation:

\[ H_{cl} + \frac{\partial \Sigma}{\partial t} = 0 \qquad \frac{\partial \rho}{\partial t} + \frac{\partial }{\partial q} \left( \rho \frac{\partial H_{cl}}{\partial p} \right) = 0 \]

This is the semiclassical limit of the Mandelung equations.

Open Science

This is a very interesting talk:

I am a bit interested in this movement. Maybe I will try to be part of it by writing on this blog.

What is this blog (good) for?

I am no stranger to blogging. Previously, I tried keeping a blog over at Wordpress.com with the goal of typing my lecture notes. Wordpress was ideal because it had LaTeX support and I was planning on typing a lot of math. Alas, the typing quickly felt to the back burner. Besides, the equations never looked that great.

Now I am returning to Blogger. MathJax lured me back. So far I have posted a few excerpts from my LaTeX files with little modification and I am pleased with how everything looks. Maybe, just maybe, I will try to post here some of the stuff that distracts me and keeps me from making progress on my research.

Propagator for harmonic potential

For the harmonic oscillator, the action looks like: