Physics 570 Assignment #3 January 28, 2006
Due Friday, February 10, 2006

1. Consider a theory of a free Dirac field of mass $m$, defined in a cubic periodic volume of size $L^3$, at (inverse) temperature $\beta$.
   1. What is the functional integral representation for the partition function? What (precisely) is the appropriate action?
   2. Show that the free energy is given by the logarithm of a functional determinant, $F = -\beta^{-1} \ln \, \det_- \, (\slash\partial + m)$, where the subscript on $\det_-$ indicates that the operator is acting on the space of antiperiodic functions.
   3. Express the derivative of the free energy with respect to mass, $\partial F / \partial m^2$, as a sum involving the eigenvalues of $\slash\partial + m$. (Find these eigenvalue explicitly.)
   4. Perform the sum over frequencies, take the infinite volume limit, and express ${\partial F / \partial m^2}$ as a spatial momentum integral with an integrand involving the usual Fermi distribution function.
   5. Integrate with respect to $m^2$ to find the free energy, and finally differentiate with respect to $\beta$ to find the expectation value of the energy. Do you find the correct result?
2. Bhabba Scattering.
   1. Derive the covariant scattering amplitude, to lowest non-trivial order, for elastic electron-positron scattering, $e^+ + e^- \to e^+ + e^-$.
   2. Evaluate the spin-averaged square of the matrix element and derive the differential cross section. Express the various invariant dot products of 4-vectors in terms of the center of mass energy $E$, the scattering angle $\theta$ and the electron rest mass, and evaluate the spin-averaged differential cross section $d\sigma/d\Omega$ (in the c.m. frame). If you wish, you may simplify the algebra by focusing on the ultrarelativistic regime, $E \gg m$.
   3. Sketch the angular distribution in the ultrarelativistic limit. How does it compare to the non-relativistic limit? What is the total scattering cross section?

The following problems are recommended, but optional. They will not be graded.

3.

Relativistic Corrections. Consider stationary solutions of the Dirac equation with a static EM field, $\left\{ \gamma^\mu ( \partial_\mu - i e A_\mu ) + m \right\} \, \psi (\undertilde x) \, e^{-i E t} = 0,$ with $E \gt 0$ and $\vert e A_0 \vert \ll m c^2$. Compute the first relativistic corrections to the spectrum of a one-electron hydrogenic atom. Specifically:

1. Solve for the lower two components of the Dirac spinor $\psi = { \psi_A \choose \psi_B }$ (in the standard representation) in terms of the upper components. Rewrite the Dirac equation as an `effective' Schrodinger equation for $\psi_A$.
2. Expand in powers of $(v^2 / c^2)$ and $\vert e A_0\vert / m c^2$, and keep the first order corrections.
3. Explain why the appropriate normalization condition on the relativistic wavefunction is $1 = \int \! d^3 x \, \psi^\dagger \psi$. Rewrite this solely in terms of the upper components $\psi_A$. Show that this normalization condition implies that the two component (`Schrodinger') spinor $\psi_S \equiv (1 + ({\undertilde {\hat p}^2} / 8 m^2 c^2)) \, \psi_A$ satisfies the usual non-relativistic normalization condition (to order $(v^2 / c^2)$).
4. Show that $\psi_S$ satisfies $\left[ {\hat {\undertilde p}^2 \over 2m} + e A^0 - {\hat {\undertilde p}^4 ... ...2} \, \nabla \cdot \undertilde E \right] \psi_S = (E - m c^2) \, \psi_S \,.$
5. Assuming that the electric field is purely radial, as in a hydrogen atom, show that the spin dependent term may be rewritten as a spin-orbit coupling (proportional to $\undertilde S \cdot \, \undertilde L$).
6. Use standard first order perturbation theory to calculate the effects of these relativistic corrections on the spectrum of the atom. Evaluate the appropriate matrix elements and express the answer as a power series in $\alpha$.
7. Which degeneracies in the non-relativistic spectrum are lifted by these corrections? What degeneracies (if any) remain?
8. How accurate are these results? What physical effects have been omitted and how large are the associated corrections? Would knowing the exact solution to the Dirac equation in a background Coulomb field improve the accuracy of the predicted energy levels?

4.

Bosonic Coherent State Functional Integrals. Standard bosonic coherent states may be defined as $\vert z> \equiv e^{-\bar z z / 2} e^{\hat a^@ z} \vert>$ where $z$ is an arbitrary complex number (and $\bar z \equiv z^\star$), $\hat a^@$ is a canonical bosonic raising operator, and $\vert>$ is a normalized harnomic oscillator ground state defined by $a \vert> = 0$.

1. Show that $\hat a \vert z> = z \vert z>$.
2. Compute the overlap $<z'\vert z>$.
3. Prove that coherent states satisfy the completeness relation $\hat 1 = \int d\bar z \, dz \> \vert z><z\vert \,,$ where $d\bar z \, dz \equiv d({\rm Re} \, z) d({\rm Im} \, z) / \pi$. [Hint: Write $\vert z>$ as a sum of normalized harmonic oscillator eigenstates, then do the integral.]
4. Prove that ${\rm Tr}\> {\scr O} = \int d\bar z \, dz \> <z\vert {\scr O} \vert z>$.
5. Consider an arbitrary Hamiltonian $\hat H = h(\hat a^@, \hat a)$ expressed as a (normal ordered) function of $\hat a$ and $\hat a^@$. By repeatedly inserting the coherent state completeness relation, derive the functional integral representation for the partition function $Z \equiv e^{-\beta H} = \int {\scr D}\bar z {\scr D}z \> e^{-S_E [\bar z, z]}$ where $S_E[\bar z, z] = \int_0^\beta d\tau \> \{ \half \bar z(\tau) {\buildrel \l... ...htarrow \over \partial_{\!\tau}} z(\tau) + h(\bar z(\tau), z(\tau)) \} \,,$ and the integral is over all complex paths which are periodic with period $\beta$.
6. By separating $z(\tau)$ into real and imaginary parts, relate this representation to the ``Hamiltonian'' path integral derived using alternating position and momentum completeness relations.

5.

Fermionic Coherent State Functional Integrals. Generalize the preceding problem to the case of fermionic theories. Let $\hat b$ and $\hat b^@$ satisfy canonical anticommutation relations. Define fermionic coherent states by the usual formula $\vert z> \equiv e^{-\bar z z / 2} e^{\hat b^@ z} \vert>$ where $z$ (and $\bar z$) are now regarded as elements of a Grassmann algebra. (Interpret Hermitian conjugation as interchanging $z$ and $\bar z$.) As usual, $\vert>$ is the state defined by $\hat b \vert> = 0$.

1. Show that $\hat b \vert z> = \vert z> z$.
2. Compute the overlap $<z'\vert z>$.
3. Using the standard rules of Grassmann integration, prove that fermionic coherent states satisfy the completeness relation $\hat 1 = \int d\bar z \, dz \> \vert z><z\vert \,.$
4. Prove that ${\rm Tr}\> {\scr O} = \int d\bar z \, dz <{-}z\vert {\scr O} \vert z>$.
5. Consider an arbitrary Hamiltonian $\hat H = h(\hat b^@, \hat b)$ expressed as a (normal ordered) function of $\hat b$ and $\hat b^@$. By repeatedly inserting the coherent state completeness relation, derive the functional integral representation for the partition function $Z \equiv {\rm Tr} \> e^{-\beta H} = \int {\scr D}\bar z {\scr D}z \> e^{-S_E [\bar z, z]}$ where $S_E[\bar z, z] = \int_0^\beta d\tau \> \{ \half \bar z(\tau) {\buildrel \l... ...htarrow \over \partial_{\!\tau}} z(\tau) + h(\bar z(\tau), z(\tau)) \} \,,$ and the integral is over all paths for which both $z(\tau)$ and $\bar z(\tau)$ are antiperiodic with period $\beta$.