Physics 52 (Classical Field Theory) Home Page, Fall 2005

Physics 52: Classical field theories (including gauge theory)


09/16/05 The talk schedule for the season has been set. If you'd like to trade off with anyone, you can arrange it among yourselves. Just let me know.


Course Information

Course Description (from the course catalog)

Schedule Times and places:


Physics 47 or consent of the instructor

Course requirements

Statement of Intellectual Responsibility: particulars for this course

How to get the most from this class:



Required (should be available at the Jeffrey Amherst bookstore): If you are considering physics graduate school, you may also wish to take a look at:
I've asked the library to order a copy of this book. It'll be placed on reserve when it arrives.

Additional useful references (if the library doesn't have them, I'll try to get them):

  • Introductory books
  • Books at about the level of this course
  • Books at above the level of this course

    Math books:

    Key derivations / chains of logic / results to commit to memory

    Mathematica Tutorials

    We will use Mathematica 5.2 at least occasionally in the homework, to obtain numerical solutions to problems that are not analytically solvable and to simplify plotting of results. If you've never used Mathematica before, or haven't used it much, the tutorials will help you get started. They were written by Professor Hilborn and revised by Rebecca Erwin '02. If you download the file and save it to the desktop with a .nb suffix in the name, your computer will recognize it as a Mathematica notebook and will start up Mathematica automatically when you double-click on the icon, provided you have Mathematica installed. Mathematica is installed on lots of the college's public machines, including on the computers in the Physics Department computer lab. Alternately, you can pay the $140 or so to buy the student version.

    Lecture Schedule
    Week Notes Hmwk
    1. September 5 Cutting deals...

    Sept 7: Introduction to the course
    General course information. Coulomb's law. Gauss's law.

    Read: Review Griffiths E&M

    Problems: Griffiths E&M: 1.60, 1.61, 1.62, 2.47, 2.49, 3.14 (just a few review problems from Griffith to warm you up)
    [Due Sept. 15]

    2. September 12 E&M Redux: Electrostatics and Magnetostatics

    Sept 12: Electrostatics
    Differential form of Gauss's law. Electrostatic potential. Poisson's equation. Electrostatic field energy. Magnetostatics: Charge conservation. Lorentz force law. Biot-Savart law. Magnetic Gauss's law.

    Sept. 14: Magnetostatics
    Ampere's law (for magnetostatics) from Biot-Savart law. Analogs between electrostatics and magnetostatics. Vector potentials: Ampere's law in terms of vector potentials. Gauge transformations. Analog of Poisson's equation for magnetostatics. Electrodynamics: Local conservation of charge. Lenz's law and Faraday's law.

    Sept. 14 (3:30-4:30): Talks

    Sarang Gopalakrishnan:
    Michael Foss-Feig:
    A guide to Local Swimming Areas, and also a short bit of physics on Molecular Magnets

    Read: Review Griffiths E&M
    Problems: Griffiths: 5.38, 5.43, 5.51, 7.50, 7.59, 7.60, 8.12, 9.33
    [Due Sept. 22]

    3. September 19 E&M Redux II: Electrodynamics / Special Relativity

    Sept 19: Electrodynamics
    Differential form of Faraday's law. Magnetic field energy. Maxwell-Ampere law. Full set of Maxwell's equations. Electromagnetic plane waves: form of the solution, dispersion relation, phase velocity. Maxwell's equations in terms of vector potentials: Redefine scalar potential for electrodynamics. Express E and B in terms of scalar and vector potential using magnetic gauss and faraday. Electric gauss and Maxwell-Ampere in terms of potentials (looks ugly).

    Sept. 21: Gauge invariance / Conservation Laws / Special Relativity
    Gauge invariance of maxwell eqns. (expressed in terms of potential). Coulomb gauge, Lorentz gauge, and their virtues. Conservation laws: Poynting's theorem and the conservation of energy. Momentum conservation and the maxwell stress tensor. Special relativity: the fundamental postulates.

    Sept. 21 (3:30-4:30): Talks

    Speaker: Jason Merrill
    The maser as a reversible heat engine
    American Journal of Physics, Vol. 73, No. 1, pp. 63-68, January 2005
    Jason's slides can be found at: Jason's slides
    Slide system S5 is at: S5
    Provocateur: Mike Borkowski

    Speaker: David Stein
    The charge distribution on a conductor for non-Coulombic potentials
    American Journal of Physics, Vol. 69, No. 4, pp. 435-440, April 2001
    Provocateur: Michael Foss-Feig

    Primary reading:
    Landau, Chap. 1, and start Chap. 2.
    Some suggested supplementary readings:
    Griffiths, Chap. 12,
    Jackson, Chap. 11.
    [The supplementary reading aren't necessary, but they may provide a different perspective on the material than Landau's or my own.]

    Problems: By popular demand, I have taken these from Jackson (3rd ed.). Work problems 11.4, 11.5, 11.6, 11.11.
    (I'll LaTeX these up if I can. For now, help yourself to photocopies of the problems from Jackson, Chap. 11 which are are available in the blue box outside my office. Also, the book is on reserve at the science library.)
    [Due Oct. 6]

    4. September 26 Special Relativity

    Sept. 26: Invariant Intervals
    Two postulates of special relativity. Consequence: loss of simultaneity. Define the interval in Minkowski spacetime as that which is null when events are connected by light rays. Show that this interval is invariant under LT. Example: examine case in which two events happening at the same place in some frame (timelike separation).

    Sept. 28: Lorentz transformations
    Spacelike interval implies there's a frame in which events are simultaneous, and vice versa. Light cone divides Mink. space into three regions: absolute past/future, elsewhere, a light cone itself. Proper time is time clock reads in its rest frame. Outline of twin paradox. Derive explicit form of most general linear transf. that leaves interval invariant (Lorentz transf.) in terms of rapidity. Observe: form is reminiscent of rotation. Obtain explicit form of LT in terms of v. Show primed axes on spacetime diagram.

    Sept. 28 (3:30-4:30): Talks

    Speaker: Mike Borkowski
    Feynman's scandalous proof of Maxwell's equations
    (based on AJP March 1990 Volume 58, Issue 3, pp. 209-211)
    Provocateur: Jason Merrill

    Speaker: Ben Heidenreich
    Symmetry and Magnetoelectric Effects
    (based on J. App. Physics Supplement to v. 33, no. 3 (March 1962) 1126-1133, "Magnetoelectric Effects in Antiferromagnets", G. Rado and V.J. Folen)
    Provocateur: Sarang Gopalakrishnan

    Read: Start Landau, Chap. 2

    Problems: Jackson, Chap. 11: 4, 5, 6, 11 carried over from last week.
    [Due Oct. 6]

    5. October 3 4-vectors and 4-tensors

    Oct. 3: Special relativity / 4-vectors and 4-tensors
    Inverse Lorentz transformation by switching sign of v. Galilean transformations commute, Lorentz transformations don't in general. Time dilation. Length contraction. Velocity addition. Example: radiation from relativistic particle into forward cone. Geometrical underpinnings of vectors and tensors. Physical laws expressed in 4-tensor form automatically satisfy first postulate of SR. Contravariant 4-vector, 4-tensor defined. Covariant 4-vector, 4-tensor defined. Express in index notation the fact that the product of LT and inverse LT is identity.

    [References: Arfken, Chap. 2 is an OK ref. for tensors from a non-geometrical perspective. However, he is concerned primarily with tensors in Euclidean 3-space, where the transformation of interest is rotation. It turns out that there's no need for him to maintain a distinction between covariant and contravariant indices in this case, and he doesn't. For a more geometrical approach, look at the first part of most books on general relativity.]

    Oct. 5: Tensor algebra and calculus
    Mixed tensors. Tensor algebraic operations: sum, outer product, contraction (esp. to form Lorentz scalars). Parity transformation. Physical laws expressed in tensor form are manifestly covariant (satisfy 1st postulate). Quotient rule. Levi-Civita tensor (actually a pseudotensor). Dual of a tensor. Metric tensor relates covariant and contravariant vectors. Define inverse metric with raised indices. Norm of 4-vector in terms of metric. Form of metric from metric equation. Metric form in invariant in all IRFs. Proper LT have det=1, improper LT have det=-1. Define covariant derivative as derivative wrt contravariant coordinates. Contravariant derivative is analogous. 4-vector divergence and D'Alembertian of scalar field are Lorentz scalars.

    Oct. 5 (3:30-4:30): Talks

    speaker: David Schaich
    provocateur: Ben Heidenreich
    speaker: Sarang Gopalakrishnan
    provocateur: David Stein

    Read: Landau, Chap. 1 and 2
    Problems: Jackson Chap. 11: 10, 12, 13, 14, 15, 23
    [Due Oct. 13]

    6. October 12 Covariant formulation of electromagnetism

    Oct. 10: Fall break

    Oct. 12: Maxwell equations in covariant form
    Volume in Minkowski space is Lorentz scalar. Covariant continuity equation. Four-current density of a point charge. Potential Maxwell equations in covariant form (in Lorentz gauge). Argue that E and B cannot transform as vectors, must be components of tensor.

    Oct. 12 (3:30-4:30): Talks

    speaker: Michael Foss-Feig
    speaker: Jason Merrill


    6. October 17 Field transformations / Transformations as groups

    Oct. 17: Lorentz transformations of E and B fields

    Introduce field strength tensor F, express in terms of E and B. F is gauge invariant. F is a true tensor, second rank antisymmetric tensor. Introduce *F, the dual of F. *F is a pseudotensor. FF, and *F*F are Lorentz scalars, *FF is a pseudoscalar. Maxwell equations in terms of F and *F. Use F to show how E and B transform under LT.

    Oct. 19: Introduction to symmetry groups

    Abstract group defined (set with four properties: closure, identity, inverse, associativity). Example: rotations in plane. Representations of groups: realizations of group multiplication rule in terms of matrices (automatically satisfies associativity), i.e. homomorphism from group onto GL(n,R) or GL(n,C) (n is the dimension of rep). Faithful rep: homomorphism is an isomorphism. Equivalent reps: related by similarity transf. Unitary reps: all D(g) are unitary. Completely reducible, reducible, and irreducible reps defined. Permutation group (to be covered in detail by MB). Unitary group. Orthogonal group. Orthogonal group elements (in defining rep.) are matrices that rotate 3-vectors. Infinitesimal generators of rotations.

    Oct. 19 (3:30-4:30): Talks

    speaker: David Stein
    speaker: Mike Borkowski


    7. October 24 Continuous spacetime transformations

    Oct. 24: Rotations

    Exponentiating the inf. generator gives finite rotation (more generally, exp. the inf. generator gives the exponential parametrization of the group elements). Lie algebra defined. Closure of group under mult. means closure of vector space spanned by Lie algebra generators under commutation. SU(2) is 2-1 covering group of SO(3). Parametrize elements of SU(2). Explicitly map SU(2) into SO(3) (show for rotations about z-axis).

    Oct. 26: D-dimensional rotations and Lorentz transformations

    2-1 relation between SU(2) and SO(3). Spinor reps of SU(2). Rotations in D-dim. Generators. Commutations relations.

    Oct. 26 (3:30-4:30): Talks

    speaker: Ben Heidenreich
    speaker: David Schaich


    8. October 31 Relativistic particle mechanics from action principles

    Oct. 31: Relativistic free particle: action

    Nov. 2: Relativistic free particle: motion, conservation laws

    Nov. 2 (3:30-4:30): Talks

    speaker: Sarang Gopalakrishnan
    speaker: Michael Foss-Feig


    9. November 7 Fields through action principles

    Nov. 7: Charged particles in external fields / a simple field theory

    Nov. 9

    Nov. 9 (3:30-4:30): Talks

    speaker: Jason Merrill
    speaker: David Stein


    10. November 14 title

    Nov. 14:

    Nov. 16:

    Nov. 16 (3:30-4:30): Talks

    speaker: Mike Borkowski
    speaker: Ben Heidenreich


    11. November 28 title

    Nov. 28:

    Nov. 30:

    Nov. 30 (3:30-4:30): Talks

    speaker: David Schaich
    speaker: Sarang Gopalakrishnan


    12. December 5 title

    Dec. 5:

    Dec. 7:

    Dec. 7 (3:30-4:30): Talks

    speaker: Michael Foss-Feig
    speaker: Jason Merrill


    13. December 12 title

    Dec. 12:

    Dec. 14:

    Dec. 14 (3:30-4:30): Talks

    speaker: David Stein
    speaker: Mike Borkowski


    Finals week Final exam: TBA


    I'll keep scheduling information on this site primarily. I may occasionally use Blackboard as well.

    Useful Links

    I'll post interesting or useful links pertinent to the course here as they I come across them. If you come across any others, please let me know.

    Interesting talks in the Five-College area:

    You should start attending the departmental colloquia early and often. They are intended primarily for you, to broaden your exposure to current physics in ways that the department faculty alone cannot. They'll give you an overview of what exciting work is going on in physics and who's doing it. In the beginning you won't always understand all of the talks, but you'll be surprised by how much you can understand even now. In addition, the colloquium food here is better than anywhere else I've ever been. Plus, I organize the colloquia, and it warms my heart to see you there.

    Area Seminars and colloquia

    Interesting and useful papers:

    Here are some papers that may be good to talk about for your Monday talks. Not all papers are necessarily appropriate (they're here primarily because the title looked to me somewhat field-theoretic or had some relevance to some sub-topic we'll cover in the course), and just because they're listed here doesn't mean you don't have to check them with me. But this list is a good place to start looking.

    Interesting and useful websites: