I decided to get back to some self-study on QFT as working toward some deep learning fun project involving QM9, and this is the first post in the series. The goal is to translate the physics, notation-wisely and conceptually, to the language of mathematics a numerical analyst such as myself is familiar with.

The main reference, is Folland’s book with a catchy title Quantum Field Theory: A tourist guide for mathematicians, which is also one of my own favorite, including the following quote

But there is another difficulty of a more cultural and linguistic nature: physics texts are usually written by physicists for physicists.

They speak a different dialect, use different notation, emphasize different points, and worry about different things than mathematicians do, and this makes their books hard for mathematicians to
read. (Physicists have exactly the same complaint about mathematics books!)

(Folland, 2008)

History

J. Schwinger formed the action principle for QFT back in 1951 in (Schwinger, 1951). People were using so-called perturbative method to perform the approximation, which reads essentially as a series expansion. Wilson developed a non-perturbative approach in (Wilson, 1974). It is now known as the LGT (lattice gauge theory), using the space-time discretization, with all the physical quantities living on the lattice, to ‘‘approximate’’ the models in QFT. For a pedagogical review article, see (Kogut, 1983) with a plethora of actual examples.

Lattice is a cubical mesh structure where the finite difference can be used to approximate the covariant derivatives. There are simplicial lattices as well. The term ‘‘lattice’’ used by physicists can be viewed as the term ‘‘mesh’’ adopted by numerical analysts.

Physics community has been studying how to compute LGT numerically, a widely-used method is Monte-Carlo simulation (e.g. (Ardill et al., 1983) (Cahill et al., 1984) (Drouffe & Moriarty, 1983) on simplicial lattices). The LGT is so powerful that it can be used to compute, or even predict,
many real life physical quantities, e.g. see (Fodor & Hoelbling, 2012) for masses of light hadrons. For an introduction, a nice reference is (Lepage, 2005).

Back then in the 80s, (Bender et al., 1983)(Bender & Sharp, 1983)(Bender et al., 1985), (Matsuyama, 1987)(Matsuyama, 1985), and (Vázquez, 1985) studied what they called ‘‘finite element’’ to solve the equations of motion from lattice quantum field theory. Even though the methods they developed all used a finite-difference type operator, still they dubbed the method as finite element. The Dirac equation is converted to a corresponding difference equation, by replacing derivatives by forward differences, while non-differentiated terms are averaged over (the covariant correction term).

In the 21st century, least-square finite elements and the multigrid solvers are studied for Dirac equation from QED in (missing reference)(Ketelsen et al., 2010). Similar finite element formulations on simplices are developed in (Christiansen & Halvorsen, 2012)(Halvorsen & Sørensen, 2012)(Halvorsen & Sørensen, 2013) Multigrid method for the linear system from lattice QCD was studied in (Frommer et al., 2014), in which the authors put a good effort in explaining the fundamentals and translating the physicist’s language to appeal the scientific computing community.

Lagrangian formalism

To work out the physics of a certain subject (or system), several things have to be specified beforehand: coordinates (usually displacement and momentum), action (integral of the Lagrangian over time), and a background gauge field. For simplicity, this note always uses geometrized units where the speed of light, as well as the Planck constant, is set to be equal to 1.

The Lagrangian allows us to find out the equations of motion (Euler-Lagrange equations) of the physical system, its symmetries and (via Noether’s theorem) the corresponding conserved quantities.

The action is a functional sometimes given in the following form:

\[S = \int^{\tau}_0 \int_{\mathbb{R}^3} \mathcal{L}(\boldsymbol{x},\dot{\boldsymbol{x}}) \,d\boldsymbol{x} dt. \tag{1}\label{eq:action}\]

It represents the integration of the difference between kinetic energy and potential energy during the time we start and stop the stopwatch. The least action principle, which is quite ‘‘metaphysical’’, states that nature takes the simplest possible configurations. It says that known a particle’s trajectory from a point $(t_0,\boldsymbol{x}_0)$ in spacetime, to another point $(\tau, \boldsymbol{\xi})$, any small perturbation lead to an increase in the action $S$.

In general, the Lagrangian (density) $\mathcal{L}$ does not have to be a function about the displacement and momentum. Instead, it can be a function of the general coordinate $\phi$:

\[S = \int^{\tau}_0 \int_{\mathbb{R}^3} \mathcal{L}(\phi,\partial_{\mu} \phi) \,d\boldsymbol{x} dt. \tag{2}\label{eq:action-4d}\]

For example, $\phi = (\theta_1,\theta_2)$ being the angles between the two rods with the vertical axis in the double pendulum system.

The Euler-Lagrange equation is obtained by perturbing the action functional using a different ‘‘path’’ in space but the starting and ending time, as well as boundary behavior in space, are unchanged. We can state the least action principle above in functional derivative:

\[\frac{\delta S}{\delta \phi} = 0. \tag{3}\label{eq:action-lst}\]

Suppose the action here we use follows from the form of \eqref{eq:action}, in a bounded domain $\Omega$ instead of $\mathbb{R}^3$, and using $\phi(\boldsymbol{x})$ and $\nabla \phi(\boldsymbol{x})$ as the input, we have:

\[\begin{aligned} & \int^{\tau}_0\int_{\Omega} \frac{\delta S}{\delta \phi}(\boldsymbol{x}) \xi(\boldsymbol{x}) \,d\boldsymbol{x} dt \\ = & \lim_{\epsilon\to 0} \frac{d}{d \epsilon}\int^{\tau}_0 \int_{\Omega} \mathcal{L}(\phi + \epsilon \xi, \nabla \phi + \epsilon \nabla \xi) \,d\boldsymbol{x} dt \\ =& \int^{\tau}_0 \int_{\Omega} \left(\frac{\partial \mathcal{L}}{\partial\phi} \xi + \frac{\partial \mathcal{L}}{\partial(\nabla \phi)}\cdot \nabla \xi \right)\,d\boldsymbol{x} dt \\ =& \int^{\tau}_0 \int_{\Omega} \left[ \frac{\partial \mathcal{L}}{\partial\phi} \xi - \nabla \cdot\Big( \frac{\partial \mathcal{L}}{\partial(\nabla \phi)}\Big) \xi\right] \,d\boldsymbol{x} dt. \end{aligned}\]

Thus \eqref{eq:action-lst} becomes

\[\frac{\partial \mathcal{L}}{\partial\phi} - \nabla \cdot\left( \frac{\partial \mathcal{L}}{\partial(\nabla \phi)}\right) = 0.\]

The four-vector version based on \eqref{eq:action-4d} is:

\[\frac{\partial \mathcal{L}}{\partial\phi} - \nabla_{\mu} \cdot\left( \frac{\partial \mathcal{L}}{\partial(\nabla_{\mu} \phi)}\right) = 0, \tag{4}\label{eq:el}\]

wherein contrary to above, the $\phi$ may depend on time $t$. Or in index notation (Einsteins notation)

\[\frac{\partial \mathcal{L}}{\partial\phi} - \sum_{\mu=0}^{3}\partial_{\mu}\left( \frac{\partial \mathcal{L}}{\partial(\partial_{\mu} \phi)}\right) = 0, \tag{5}\label{eq:el-idx}\]

where the repeated index gets summed up (we specify the summation explicitly versus the implicit summation in physics literatures).

Example 1: the Klein-Gordon equation

Consider a massive scalar field, the field strength at $x = (t,\boldsymbol{x})$ is $\phi(x)$, a complex scalar. The Lagrangian in this case, is

\[\mathcal{L} = \frac{1}{2} (\nabla_{\mu} \phi)^2 - \frac{1}{2}m^2\phi^2 = \frac{1}{2}\sum_{\mu=0}^{3} (\partial^{\mu} \phi) (\partial_{\mu} \phi) - \frac{1}{2}m^2\phi^2. \tag{6}\label{eq:kg-lg}\]

This is equivalent to

\[\mathcal{L} = \frac{1}{2} (\partial_t \phi)^2 - \frac{1}{2} |\nabla \phi|^2 - \frac{1}{2}m^2 \phi^2.\]

The Lagrangian is in the form of kinetic energy subtracting the potential energy. Plugging into Euler-Lagrange equation formula \eqref{eq:el} we have:

\[\frac{\partial \mathcal{L}}{\partial\phi} = -m^2\phi,\quad \text{and } \; \frac{\partial \mathcal{L}}{\partial(\nabla_{\mu} \phi)} = \nabla^{\mu} \phi = \partial_t \phi - \nabla\phi.\]

The E-L equation is then

\[(\nabla_{\mu}\cdot \nabla^{\mu} + m^2 )\phi = 0,\]

or using vector notation

\[\left(\partial^2_t - \Delta + m^2\right) \phi = 0.\]

The spacial variable in this equation lies in the free space $\mathbb{R}^3$, and the problem of how to impose boundary condition for computation will be discussed later.

References

  1. Folland, G. B. (2008). Quantum Field Theory: A tourist guide for mathematicians (Number 149). American Mathematical Soc.
  2. Schwinger, J. (1951). The theory of quantized fields. I. Physical Review, 82(6), 914.
  3. Wilson, K. G. (1974). Confinement of quarks. Physical Review D, 10(8), 2445.
  4. Kogut, J. B. (1983). The lattice gauge theory approach to quantum chromodynamics. Reviews of Modern Physics, 55(3), 775.
  5. Ardill, R. W. B., Clarke, J.-P., Drouffe, J.-M., & Moriarty, K. J. M. (1983). Quantum chromodynamics on a simplical lattice. Physics Letters B, 128(3), 203–206.
  6. Cahill, K., Prasad, S., & Reeder, R. (1984). Simplicial interpolations for path integrals. Physics Letters B, 149(4), 377–380.
  7. Drouffe, J. M., & Moriarty, K. J. M. (1983). Gauge theories on a simplicial lattice. Nuclear Physics B, 220(3), 253–268.
  8. Fodor, Z., & Hoelbling, C. (2012). Light hadron masses from lattice QCD. Reviews of Modern Physics, 84(2), 449.
  9. Lepage, G. P. (2005). Lattice QCD for novices. ArXiv Preprint Hep-Lat/0506036.
  10. Bender, C. M., Milton, K. A., & Sharp, D. H. (1983). Consistent formulation of fermions on a Minkowski lattice. Physical Review Letters, 51(20), 1815.
  11. Bender, C. M., & Sharp, D. H. (1983). Solution of operator field equations by the method of finite elements. Physical Review Letters, 50, 1535–1538. https://doi.org/10.1103/PhysRevLett.50.1535
  12. Bender, C. M., Milton, K. A., Sharp, D. H., Simmons Jr, L. M., & Stong, R. (1985). Discrete-time quantum mechanics. Physical Review D, 32(6), 1476.
  13. Matsuyama, T. (1987). Quantum field theory on a finite element: Self-coupled scalar field theory. Nuclear Physics B, 289, 277–300.
  14. Matsuyama, T. (1985). U(1) gauge-coupled Dirac equation and massless (QED)_2 on a finite element. Physics Letters B, 158(3), 255–263.
  15. Vázquez, L. (1985). Dirac field on a Minkowski lattice. Physical Review D, 32(8), 2066.
  16. Ketelsen, C., Manteuffel, T., McCormick, S., & Ruge, J. (2010). Finite element methods for quantum electrodynamics using a Helmholtz decomposition of the gauge field. Numerical Linear Algebra with Applications, 17(2-3), 539–556.
  17. Christiansen, S. H., & Halvorsen, T. G. (2012). A simplicial gauge theory. Journal of Mathematical Physics, 53(3), 033501.
  18. Halvorsen, T. G., & Sørensen, T. M. (2012). Simplicial gauge theory and quantum gauge theory simulation. Nuclear Physics B, 854(1), 166–183.
  19. Halvorsen, T. G., & Sørensen, T. M. (2013). Simplicial gauge theory on spacetime. Numerische Mathematik, 125(4), 733–760.
  20. Frommer, A., Kahl, K., Krieg, S., Leder, B., & Rottmann, M. (2014). Adaptive Aggregation-Based Domain Decomposition Multigrid for the Lattice Wilson–Dirac Operator. SIAM Journal on Scientific Computing, 36(4), A1581–A1608.

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