TranslationInvariantOperators

In quantum field theory, the role of symmetries is highlighted by Noether's theorem, which associates a conserved current to every continuous symmetry of the action.

For instance, for a translation by a small constant four-vector $\epsilon$, the change in the action can be written as:

$$\Delta S = \int \partial_\mu J^\mu d^4x$$

where $J^\mu$ is the Noether current, which for translations is just the energy-momentum tensor $T^{\mu\nu}$:

$$J^\mu = i T^{\mu\nu} \epsilon_\nu$$

This highlights the deep connection between symmetries and conserved quantities in quantum field theory.

Yang-Mills theories are a particular type of quantum field theory where the fields are vectors in some Lie algebra, and the action is invariant with respect to the associated Lie group of continuous gauge transformations.

For a Yang-Mills quantum field theory, the energy-momentum tensor is much more complicated due to the non-trivial self-interactions of the fields.

For a Yang-Mills theory with no matter fields, the energy-momentum tensor can be written in terms of the field strength $F^a_{\mu\nu}$ for the $a$-th component of the gauge field:

$$T^{\mu\nu} = - F^{a \mu \lambda} F^{a \nu}{\ \ \lambda} + \frac{1}{4}g^{\mu\nu} F^{a \lambda \rho} F^{a}{\ \ \lambda \rho}$$

In these expressions, the indices $a$ label elements of the Lie algebra, and the indices $\mu$, $\nu$, $\rho$, $\sigma$ are spacetime indices.

The invariance of the Yang-Mills action under translations (shifts) leads, via Noether's theorem, to the conservation of the energy-momentum tensor:

$$\partial_\mu T^{\mu\nu} = 0$$

which expresses the conservation of energy and momentum in a Yang-Mills theory.

The Wightman axioms, on the other hand, postulate certain conditions that a quantum field theory should satisfy in order to give rise to a well-defined particle interpretation. In terms of the momentum operator $P^\nu$, defined as the integral of the spatial components of the energy-momentum tensor over all space:

$$P^\nu = \int_{\mathbb{R}^3} T^{0\nu} dx$$

the axiom related to the conservation of momentum states that the vacuum is invariant under momentum translations:

$$P^\nu |0\rangle = 0$$

This ties together the translation-invariance symmetry with the conservation of momentum and physical interpretation of the theory via the vacuum state.

The $T^{0\nu}$ notation is used specifically to denote the components of the energy-momentum tensor that are integrated to get the momentum operator $P^\nu$ in quantum field theory.

The superscript $0$ refers to the time component, and $\nu$ is a spacetime index that can take values from $0$ to $3$ in a 4-dimensional spacetime. So, $T^{0\nu}$ represents the components of the energy-momentum tensor involving time and one of the spacetime dimensions.

Indeed, $T^{\mu\nu}$ where $\mu$ and $\nu$ are spacetime indices each ranging over $0$ to $3$ represent general elements of the energy-momentum tensor and $T^{0\nu}$ arises in the specific definition of the 4-momentum operator $P^\nu$.