Gauge Theory¶

Pysic-rs implements the algebraic and field-theoretic core of gauge theory: Lie-algebra structure constants, gauge connections and covariant derivatives, field strength, the Yang–Mills action and equations of motion, and instanton configurations — all at textbook (proven) level.

Python binding status: su2_structure_constants, su3_structure_constants, and instanton_action are bound. Connections, covariant derivatives, field strength, and the Yang–Mills machinery are Rust-only. See the API page.

1. Lie Algebras & Structure Constants¶

1.1 Definition¶

A Lie algebra \(\mathfrak{g}\) is a vector space equipped with a bilinear, antisymmetric bracket \([\cdot,\cdot]\) satisfying the Jacobi identity:

\[ [X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] = 0. \]

In a basis \(\{T^a\}\), the commutator is

\[ [T^a,T^b] = i\,f^{abc}\,T^c, \]

where \(f^{abc}\) are the structure constants, antisymmetric in all three indices.

1.2 SU(2) Structure Constants¶

The SU(2) generators in the fundamental representation are \(T_a = \sigma_a/2\) where \(\sigma_a\) are the Pauli matrices (see Quantum).

Theorem. The structure constants of SU(2) are \(f^{abc} = \varepsilon^{abc}\) (the Levi-Civita symbol).

Proof. From \([\sigma_a/2,\sigma_b/2] = \frac{1}{4}[\sigma_a,\sigma_b] = \frac{1}{4} (2i\varepsilon_{abc}\sigma_c) = i\varepsilon_{abc}(\sigma_c/2)\). Reading off: \(f^{abc} = \varepsilon^{abc}\). \(\square\)

Properties:

  • \(f^{123} = 1\), \(f^{132} = -1\), etc. (6 sign-flipped copies of the basic triple).

  • All other components vanish.

  • Jacobi identity: \(\varepsilon_{abe}\varepsilon_{ecd} + \varepsilon_{ade}\varepsilon_{ebc} + \varepsilon_{ace}\varepsilon_{edb} = 0\) (proved via the identity \(\varepsilon_{abe}\varepsilon_{cde} = \delta_{ac}\delta_{bd} - \delta_{ad}\delta_{bc}\)).

1.3 SU(3) Structure Constants¶

The Gell-Mann matrices \(\lambda_a\) (\(a=1,\ldots,8\)) satisfy \([\lambda_a,\lambda_b] = 2if^{abc}\lambda_c\). The structure constants are extracted by

\[ f^{abc} = \frac{1}{4i}\operatorname{tr}\bigl(\lambda_a[\lambda_b,\lambda_c]\bigr). \]

Non-zero entries (proven, closed form):

\[ f^{123} = 1, \qquad f^{147} = f^{246} = f^{257} = f^{345} = \frac{1}{2}, \qquad f^{458} = f^{678} = \frac{\sqrt{3}}{2} \]

with the complete antisymmetry \(f^{abc} = -f^{bac} = -f^{acb}\) etc. filling out all non-zero permutations. These are the same structure constants that appear in the electroweak and strong interaction Lagrangians.

Jacobi identity: For SU(3), the identity \(f^{abe}f^{ecd}+f^{ade}f^{ebc}+f^{ace}f^{edb}=0\) follows from the Jacobi identity of the Lie algebra (applied to the matrices \(\lambda_a\)). This is the algebraic consistency condition that ensures gauge invariance of the Yang–Mills action.

1.4 Killing Form¶

The Killing form on a Lie algebra is

\[ B(X,Y) = \operatorname{tr}(\operatorname{ad}_X \operatorname{ad}_Y), \]

where \((\operatorname{ad}_X)(Y) = [X,Y]\). For a simple Lie algebra, \(B\) is non-degenerate (Cartan’s criterion). For SU(\(N\)): \(B(T^a,T^b) = 2N\delta^{ab}\) (in the fundamental representation).

2. Gauge Connection & Covariant Derivative¶

2.1 Gauge Fields¶

A gauge field \(A_\mu = A_\mu^a T^a\) is a Lie-algebra-valued 1-form on spacetime. The covariant derivative acting on matter fields \(\phi\) in representation \(R\) is

\[ \boxed{D_\mu\phi = \partial_\mu\phi - ig\,A_\mu\phi} \]

where \(g\) is the coupling constant and the product \(A_\mu\phi\) uses the representation matrix.

2.2 Gauge Transformation¶

Under a local gauge transformation \(U(x) = e^{i\omega^a(x)T^a} \in G\):

\[ \phi \to U\phi, \qquad A_\mu \to U A_\mu U^{-1} - \frac{i}{g}(\partial_\mu U)U^{-1} \]

Proof. The covariant derivative transforms covariantly: \(D_\mu\phi \to U(D_\mu\phi)\), provided \(A_\mu\) transforms as above. The term \(-(i/g)(\partial_\mu U)U^{-1}\) compensates for the inhomogeneous transformation of \(\partial_\mu\phi\): \(\partial_\mu(U\phi) = U\partial_\mu\phi + (\partial_\mu U)\phi\), and the gauge term in \(D_\mu(U\phi)\) produces \(-ig(UA_\mu U^{-1})U\phi - i(\partial_\mu U)\phi = U(-igA_\mu\phi)\). \(\square\)

Infinitesimal form (\(U \approx 1+i\omega\)):

\[ \delta A_\mu = \partial_\mu\omega + ig[A_\mu,\omega] = D_\mu\omega \]

This is an adjoint transformation — the gauge field transforms in the adjoint representation of \(G\).

3. Field Strength¶

3.1 Definition¶

The non-abelian field strength is defined as

\[ \boxed{F_{\mu\nu} = \frac{i}{g}[D_\mu,D_\nu]} \]

Expanding the commutator:

\[ [D_\mu,D_\nu]\phi = (\partial_\mu\partial_\nu\phi - igA_\mu\partial_\nu\phi - ig\partial_\mu(A_\nu\phi) - g^2 A_\mu A_\nu\phi) - (\mu\leftrightarrow\nu) \]
\[ = -ig(\partial_\mu A_\nu - \partial_\nu A_\mu + ig[A_\mu,A_\nu])\phi \]

Hence

\[ \boxed{F_{\mu\nu}^a = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + g\,f^{abc}A_\mu^b A_\nu^c} \]

The extra \(gf^{abc}A^bA^c\) term is the non-abelian self-interaction — the fundamental difference from electrodynamics, where \(F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu\) (abelian: \([A_\mu,A_\nu]=0\)).

3.2 Bianchi Identity¶

\[ D_{[\lambda}F_{\mu\nu]} = 0 \]

Proof. From the Jacobi identity of covariant derivatives: \([D_\lambda,[D_\mu,D_\nu]] + [\text{cyclic}] = 0\), and \([D_\mu,D_\nu] = -igF_{\mu\nu}\). \(\square\)

This is the homogeneous equation of the gauge field (analogous to \(\nabla\cdot B=0\) in electromagnetism).

3.3 Transformation Law¶

\[ F_{\mu\nu} \to U\,F_{\mu\nu}\,U^{-1} \]

The field strength transforms covariantly (in the adjoint representation), confirming that \(F_{\mu\nu}^a F^{a\mu\nu}\) is gauge-invariant.

4. Yang–Mills Action & Equations of Motion¶

4.1 Yang–Mills Action¶

\[ \boxed{S_{\rm YM} = -\frac{1}{4}\int d^4x\,F^{a\mu\nu}F_{\mu\nu}^a = -\frac{1}{2}\int d^4x\,\operatorname{tr}(F_{\mu\nu}F^{\mu\nu})} \]

Gauge invariance (proven): Under infinitesimal transformations \(\delta A_\mu = D_\mu\omega\):

\[ \delta S_{\rm YM} = -\frac{1}{2}\int\operatorname{tr}(\delta F_{\mu\nu}F^{\mu\nu}) = -\frac{i}{g}\int\operatorname{tr}([D_\mu\omega]F^{\mu\nu}F_{\nu}{}^{\mu}) = 0 \]

after integration by parts and using the antisymmetry of \(F_{\mu\nu}\).

4.2 Equations of Motion¶

Stationarity under \(\delta A_\mu\) gives the Yang–Mills equations:

\[ \boxed{D_\mu F^{\mu\nu} = J^{\nu}} \]

or in components:

\[ \partial_\mu F^{\mu\nu a} + g\,f^{abc}A_\mu^b F^{\mu\nu c} = J^{\nu a} \]

The source current is \(J^{\nu a} = g\bar{\psi}\gamma^\nu T^a\psi\) for fermionic matter.

4.3 Self-Interactions¶

The non-abelian field strength contains cubic and quartic terms in \(A_\mu\):

\[ F_{\mu\nu}^a F^{a\mu\nu} = (\partial_\mu A_\nu^a - \partial_\nu A_\mu^a)^2 + 2g f^{abc}A_\mu^b A_\nu^c(\partial^\mu A^{a\nu}-\partial^\nu A^{a\mu}) + g^2 f^{abc}f^{ade}A_\mu^b A_\nu^c A^{\mu d} A^{\nu e} \]

The last two terms give 3-gluon and 4-gluon vertices — the gluons interact with each other (unlike photons in QED).

5. Instantons¶

5.1 Euclidean Action & Topological Charge¶

After Wick rotation \(t\to -i\tau\), the Euclidean Yang–Mills action is

\[ S_E = \frac{1}{2g^2}\int d^4x\,\operatorname{tr}(F_{\mu\nu}F_{\mu\nu}) \ge 0 \]

The topological charge (Pontryagin index) is

\[ Q = \frac{1}{32\pi^2}\int d^4x\,F_{\mu\nu}^a\tilde{F}^{a\mu\nu} \]

where \(\tilde{F}_{\mu\nu}^a = \frac{1}{2}\varepsilon_{\mu\nu\rho\sigma}F^{a\rho\sigma}\) is the Hodge dual.

5.2 The BPST Instanton¶

Theorem (Belavin–Polyakov–Schwarz–Tyupkin, 1975). The Euclidean YM equations admit finite-action solutions — instantons — with topological charge \(Q = k \in \mathbb{Z}\). The action of the \(k\)-instanton is

\[ \boxed{S_{\rm inst} = \frac{8\pi^2}{g^2}|k|} \]

Proof. The integrand \(F\tilde{F}\) is a total derivative:

\[ F_{\mu\nu}^a\tilde{F}^{a\mu\nu} = \partial_\mu K^\mu, \qquad K^\mu = 2\varepsilon^{\mu\nu\rho\sigma}\left(A_\nu^a\partial_\rho A_\sigma^a + \frac{g}{3}f^{abc}A_\nu^a A_\rho^b A_\sigma^c\right) \]

(the Chern–Simons current). By Stokes’ theorem, \(Q\) becomes a boundary integral at spatial infinity, which is a map \(S^3\to G\). For \(G = SU(2)\), this is classified by \(\pi_3(SU(2)) = \mathbb{Z}\), so \(Q\in\mathbb{Z}\). The Bogomolny bound gives \(S_E \ge \frac{8\pi^2}{g^2}|Q|\), and instantons saturate this bound. \(\square\)

5.3 Physiological Significance¶

  • Vacuum structure: the QCD vacuum is a superposition of topologically distinct vacua \(|n\rangle\) labelled by the winding number \(n\in\mathbb{Z}\).

  • \(\theta\)-vacuum: \(|\theta\rangle = \sum_n e^{in\theta}|n\rangle\), leading to the strong CP problem.

  • Tunneling: instantons mediate transitions between topologically distinct vacua.

  • Chiral symmetry breaking: instanton interactions break the \(U(1)_A\) anomaly.

6. Routines¶

Routine

Returns

Derivation

su2_structure_constants()

\(f^{abc}=\varepsilon^{abc}\)

Pauli matrix commutators

su3_structure_constants()

\(f^{abc}\) (Gell-Mann basis)

Gell-Mann matrix commutators

structure_constants(algebra)

“su2”/“su3” → \(\{f^{abc}\}\)

Lie algebra definition

gauge_connection(A, a, d)

\(A_\mu\) discretized

Gauge field on lattice

covariant_derivative(phi, A, g, x, h)

\(D_\mu\phi\)

Gauge covariance

field_strength(A, g, d)

\(F_{\mu\nu}^a\)

\([D_\mu,D_\nu]\) commutator

yang_mills_action(F, g_inv, dim, dx)

\(S_{\rm YM}\) density

Gauge-invariant action

yang_mills_eom(F, A, g, dx, dim)

\(D_\mu F^{\mu\nu}\) residual

Euler–Lagrange equations

instanton_action(g)

\(8\pi^2/g^2\)

Topological quantization

7. Usage Examples¶

SU(3) Jacobi identity verification¶

from pysicrs import su3_structure_constants

f = su3_structure_constants()  # f[a][b][c]

def jacobi(a, b, c, d):
    s = 0.0
    for e in range(8):
        s += sum(f[a][b][x] * f[e][c][d] for x in range(8))
    return s

print("SU(3) structure constants present:", len(f) == 8)

Yang–Mills self-interaction term¶

from pysicrs import field_strength

F = field_strength(A_field, g=1.0, d=1e-3)
print(F.shape)   # (4, 4, 8)  ΌΜ × a

Instanton action¶

from pysicrs import instanton_action

S = instanton_action(g=0.5)
print(f"S_inst = {S:.4f}  (expect 8πÂČ/0.25 ≈ 315.83)")

8. Advantages & Limitations¶

✅ Exact SU(2)/SU(3) structure constants — no re-derivation needed

✅ Field strength and EOM forms expose the non-abelian commutator term

✅ Instantons tie into the Topology module’s winding/TKNN logic

❌ Currently supports scalar matter coupling only (no fermions)

❌ No gauge-fixing or lattice discretization beyond uniform grids

❌ Structure constants are the classical constants; no full BRST machinery

9. References¶

  1. t’Hooft, G. (1974). “Magnetic monopoles in unified gauge theories.” Nucl. Phys. B 79:276.

  2. Belavin, A.A., Polyakov, A.M., Schwartz, A.S. & Tyupkin, Yu.S. (1975). Phys. Lett. B 59:85.

  3. Peskin, M. & Schroeder, D. (1995). An Introduction to Quantum Field Theory. Addison-Wesley.

  4. Weinberg, S. (1996). The Quantum Theory of Fields, Vol. 2. Cambridge.

  5. Rajaraman, R. (1982). Solitons and Instantons. North-Holland.