Yang-Mills Theory and Principle of Representation Invariance (PRI)

The objectives of this post are 1) to briefly introduce the basic formulation of the {SU(N)} gauge theory, and 2) to postulate the principle of representation invariance (PRI):

Principle of Representation Invariance (PRI) [Ma-Wang, 2012] An {SU(N)} gauge theory must be invariant under the representation transformations (11). Namely, the Yang-Mills action (5) of the gauge fields is invariant and the corresponding gauge field equations (12) and (13) are covariant under the representation transformations (11).

As we shall see, PRI is basic logic requirement for an SU(N) gauge theory and was first postulated by the authors in Article [4]. This principle has profound physical implications, which we shall explore in the next few posts. Of course, readers are invited to read Articles [3, 4, 11] for more details.

1. Gauge Fields

The search for laws of quantum systems has been a fascinating endeavor for mankind. With the development of quantum mechanics in the first half of last century, we understand that individual particles are well described by the (relativistic) Dirac equations for fermions, the (relativistic) Klein-Gordon equations for bosons, and in the low velocity case by the Schrödinger equations.

Particles interact with each other, and fundamental interactions of Nature (gravity, electromagnetism, the weak and the strong) are responsible for the structure and stability of the Universe we live in.

As mentioned in the previous post, it was to Albert Einstein’s credit and great vision that the search for laws of Nature can be achieved by discovering fundamental symmetries of Nature. Through the pioneering work of Herman Weyl, Chen-Ning Yang and Robert Mills, as well as the development, both experimental and theoretical, of standard model in particle physics, we know now that gauge symmetry provides such a fundamental symmetry of Nature.

An {SU(N)} gauge theory is a theory describing an {N} particle system, consisting of

  • {N} wave functions:

    \displaystyle \Psi =(\psi_1,\cdots ,\psi_N)^T,

  • {K \stackrel{\text{def}}{=} N^2-1} gauge fields {G^a_\mu}:

    \displaystyle G_\mu^a=(G_0^a, G_1^a, G^a_2, G^a_3)\ \ \ \ \text{ for } a=1, \cdots, K,

    each of which is a 4D vector field.

The {N} wave functions represent the {N}-particles, and the {K} gauge fields stand for the interacting potentials between these {N} particles. We consider here only the case where the {N} particles are fermions, which obey the Dirac equations. As these particles are interacting to each other, the dynamical laws of these particles are affected by the mutual dynamic interactions. The key issue is then to search for the associated interaction Lagrangian action, which is called the Yang-Mills action.

2. Principle of Gauge Invariance

In the previous post, we have demonstrated that the Lagrangian action for gravity, the Einstein-Hilbert functional, is dictated by the invariance under general coordinate transformations, the principle of general relativity.

Motivated in part by the early work of Herman Weyl, Erwin Schrödinger, Yang-Mills, due to the dynamic interactions of the {N} particles, one needs to replace the partial differential operator {\partial_\mu}, in the Dirac equations, by the differential operator {D_\mu}:

\displaystyle D_{\mu}=\partial_{\mu}+igG^k_{\mu}\tau_k, \ \ \ \ \ (1)

where {\tau_k\ (1\leq k \leq K)} are a set of {SU(N)} generators such that each element {\Omega \in SU(N)} is represented by {\Omega=e^{i\theta^k\tau_k}}, with {\theta^k} being parameters. It is now classical to show that covariance of the Dirac equations

\displaystyle \left[ i\gamma^{\mu}D_{\mu} - m \right] \Psi =0 \ \ \ \ \ (2)

implies that the {SU(N)} gauge transformation take the following form:

\displaystyle (\tilde{\Psi}, \ \ \tilde{G}^a_{\mu}\tau_a, \ \ \tilde m) =\left( \Omega\Psi, \ \ G^a_{\mu}\Omega\tau_a\Omega^{-1}+\frac{i}{g}(\partial_{\mu}\Omega)\Omega^{-1}, \ \ \Omega m\Omega^{-1} \right), \ \ \ \ \ (3)

for any

\displaystyle \Omega=e^{i\theta^k(x) \tau_k}\in SU(N), \ \ \ \ \ (4)

where {g} is a coupling constant, and {m} is the mass matrix.

Then the gauge symmetry, also called the principle of gauge invariance, can be stated now as follows:

Principle of Gauge Invariance. The electromagnetic, the weak, and the strong interactions obey gauge invariance. Namely, the Dirac or Klein-Gordon dynamical equations involved in the three interactions are gauge covariant and the actions of the interaction fields are gauge invariant under the gauge transformations (3).

Physically, gauge invariance refers to the conservation of certain quantum property of the underlying interaction. In other words, such quantum property of the {N} particles cannot be distinguished for the interaction, and consequently, the energy contribution of these {N} particles associated with the interaction is invariant under the general {SU(N)} phase (gauge) transformations.

The number {N=2} for the {SU(N)} gauge theory used by Chen-Ning Yang and Robert Mills (1954) represents 2-components of the isotopic spin for nucleon-nucleon interactions. In the classical quantum chromodynamics (QCD), {N=3} in the {SU(3)} gauge symmetry is associated with the three-color quantum number. Due to the representation invariance (PRI) to be introduced below, the coupling constant {g} in (1) and (3) represents the interaction charge that each of the {N} particle carries, and the gauge symmetry amounts to saying that one cannot distinguish the energy contribution of the {N} particles for interactions caused by the interaction charge {g}. In particular, in our theory, {N} represents the number of particles carrying the same charge {g} in the system; see [11].

3. Gauge Invariance Dictates Yang-Mills Action

The Dirac equations (2) are the dynamic equations for the {N} particle fields. The field equations involving the gauge fields {G^a_{\mu}} are determined by the the corresponding Yang-Mills action. We now show that the following Yang-Mills action is uniquely determined by both the gauge invariance and the Lorentz invariance, together with simplicity of laws of nature: 

\displaystyle L_{YM}=\int_{M^4}\left[-\frac{1}{4}\mathcal{G}_{ab}g^{\mu\alpha}g^{\nu\beta}F^a_{\mu\nu}F^b_{\alpha\beta} +\bar{\Psi}\left(i\gamma^{\mu}D_{\mu}- m\right)\Psi\right]dx, \ \ \ \ \ (5)

where {\bar{\Psi}=\Psi^{\dag}\gamma^0}, {\Psi^\dagger= (\Psi^\ast)^T} is the transpose conjugate of {\Psi}, {D_{\mu}} is covariant derivative defined by (1), {F^a_{\mu\nu}} stands for the curvature tensor associated with {D_\mu}:

\displaystyle F^a_{\mu\nu}=\partial_{\mu}G^a_{\nu}-\partial_{\nu}G^a_{\mu}+g\lambda^a_{bc}G^b_{\mu}G^c_{\nu}. \ \ \ \ \ (6)

The term {\{\mathcal{G}_{ab}\}} can be regarded as a Riemannian metric of {SU(N)}, defined by

\displaystyle \mathcal{G}_{ab}=\frac{1}{2}{\rm tr}(\tau_a\tau^{\dag}_b)=\frac{1}{4N}\lambda^c_{ad}\lambda^d_{cb}, \ \ \ \ \ (7)

where {\lambda^c_{ab}} are the structure constants of {\{\tau_a \ | \ a=1, \cdots, N^2-1\}}: {[\tau_a, \tau_b]=i \lambda^c_{ab} \tau_c.}

The Yang-Mills action can be derived in two steps as follows.

Step 1. The second part on the right-hand side of (5) is the action for the Dirac equations, which is both Lorentz and gauge invariant.

Step 2. From the mathematical point of view, {D_\mu} is the covariant derivative on the complex bundle of Dirac spinor spaces, denoted by {M \otimes_p (\mathbb C^4)^N}, where {M} is the 4D space-time manifold with the Minkowski metric {\{g_{\mu\nu}\}}. Nontrivial actions must then be constructed through the corresponding curvature. Direct calculation shows that

\displaystyle \frac{i}{g}\left[D_{\mu},D_{\nu}\right] = F_{\mu \nu}^a \tau_a, \ \ \ \ \ (8)

which justifies the definition of the curvature tensor {F^a_{\mu\nu}}. It is clear to see that {F_{\mu\nu}^a} transforms in the following fashion under the gauge transformation:

\displaystyle \tilde{F}^a_{\mu\nu}\tau_a=F^a_{\mu\nu}\Omega\tau_a\Omega^{-1}. \ \ \ \ \ (9)

As {\mu ,\nu} are the indices of 4-D Lorentz tensors, and Lorentz invariants can be constructed through contractions, we can show that the following density

\displaystyle \mathcal{G}_{ab}F^a_{\mu\nu}F^{\mu\nu b}=\mathcal{G}_{ab}g^{\mu\alpha}g^{\nu\beta}F^a_{\mu\nu}F^b_{\alpha\beta} \ \ \ \ \ (10)

is invariant under both the Lorentz transformations and the {SU(N)} gauge transformations, and the simplest form, which possesses these properties. Consequently, the Yang-Mills action is uniquely determined.

4. Principle of Representation Invariance (PRI)

In (4), an element {\Omega =e^{i\theta^a\tau_a} \in SU(N)} is expressed in terms a set of generators {\{\tau_a \ | \ a=1, \cdots, N^2-1\}}, which form a basis of the following set of Hermitian traceless {N \times N} complex matrices:

\displaystyle \mathcal H=\{ N \times N \text{ complex matrix } \tau \ | \ \text{Tr} \tau=0, \tau^{-1}= \tau^\dagger \}.

Basic logic dictates that one can consider different set of representation generators, and the resulting gauge theory should be invariant. This amounts to saying that we can take the following linear transformations of the generator bases:

\displaystyle \tilde{\tau}_a=x^b_a\tau_b, \ \ \ \ \ (11)

which we call representation transformations, where {X=(x^b_a)} are non-degenerate {K}-th order matrices. We can then define naturally {SU(N)} tensors under the transformations (11), and the following statements are clearly true:

  • {\theta^a,A^a_{\mu}}, and {\lambda^c_{ab}} are {SU(N)}-tensors.
  • {\mathcal{G}_{ab}=\frac{1}{4N} \lambda^c_{ad}\lambda^d_{cb} =\frac12 \text{Tr}(\tau_a \tau_b^\dagger)} is a symmetric positive definite 2nd-order covariant {SU(N)}-tensor, which can be regarded as a Riemannian metric on {SU(N)}.
  • The Yang-Mills action {L_{YM}} in (5) is {SU(N)} representation invariant under the representation transformations (11).
  • By PID, the gauge field equations are given by

    \displaystyle \mathcal{G}_{ab} \left[ \partial^{\nu}F^b_{\nu\mu} - g \lambda^{b}_{cd} g^{\alpha \beta}F^c_{\alpha\mu}A^d_{\beta}\right] - g \bar{\Psi} \gamma_{\mu}\tau_a \Psi = (\partial_\mu + \alpha_b A^b_\mu) \phi_a, \ \ \ \ \ (12)

    supplemented with the Dirac equations:

    \displaystyle (i\gamma^{\mu}D_{\mu}- m)\Psi =0 \ \ \ \ \ (13)

    Both equations are representation covariant under the representation transformations (11). Here the right-hand side terms in the field equations (12) are due to PID.

It is now natural for us to postulate the principle of representation invariance (PRI), as stated in beginning of this post. Again, PRI was first postulated by the authors in 2012 in [4], and in the next few posts, we shall see that PRI leads to a number of profound physical implications. One such implication is the introduction of charges for all interactions, and the other is that PRI sets certain prohibitions for establishing a theory unified different interactions.

Also, with PRI, we are able to derive for the first time both strong and weak interaction potentials, leading to explanations for such phenomena as quark confinement and asymptotic freedom. Again, we shall explore these implications in the next few posts.

Tian Ma & Shouhong Wang

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