Ward-Takahashi Identity from U(1) Symmetry

The Quantum Fate of a Conserved Current

In classical field theory, the global U(1) symmetry $\psi \to e^{iq\alpha }\psi$ leads to a conserved Noether current, $J^{\mu }=e\bar{\psi }{\gamma }^{\mu }\psi$, which satisfies the conservation equation ${\partial }_{\mu }{J}^{\mu }=0$. A natural question arises: what is the status of this equation in the quantum theory? Does the operator equation ${\partial }_{\mu }{\hat{J}}^{\mu }(x)=0$ hold true?

The answer is subtle and reveals a key feature of quantum field theory. The conservation law does not hold as a simple operator identity in all contexts. Instead, its consequences manifest when we compute correlation functions, which involve time-ordered products of operators. The derivative ${\partial }_{\mu }$ acting on a time-ordered product does not simply pass through to the operator inside. The time derivative ${\partial }_0$ fails to commute with the time-ordering symbol $T$, giving rise to extra terms when spacetime points coincide.

This leads to the fundamental insight: the divergence of the current operator within a time-ordered product is zero everywhere except at the spacetime locations of the other charged field operators in the correlator. At these points, we get "contact terms" proportional to Dirac delta functions.

This reading from Vadim Kaplunovsky's notes provides an excellent physical derivation of this result, starting directly from the operator statement of current conservation and the properties of the time-ordering operator.

The result you have just studied is the Ward-Takahashi identity in coordinate space:

Here, ${\hat{\varphi }}_j(x_j)$ are any set of charged field operators, and $q_j$ is the charge of the particle associated with field ${\hat{\varphi }}_j$. This equation beautifully expresses that charge is conserved, but in the quantum context, this conservation law manifests as a precise relationship between different correlation functions.

The Path Integral Derivation

Another powerful way to derive this identity is through the path integral formalism, a technique central to your QFT studies. The logic is elegant: we start with the path integral for a correlation function, which is invariant under a change of integration variables. We then perform a specific change of variables corresponding to a position-dependent U(1) phase rotation, $\psi (x)\to e^{iq\alpha (x)}\psi (x)$. Since this is not a symmetry of the action, the action changes by a known amount. Requiring the overall path integral to be unchanged leads directly to the Ward-Takahashi identity.

The following video provides a concise derivation using this method.

The Vertex Identity in QED

While the general coordinate-space identity is fundamental, its most celebrated application in QED is a specific relation in momentum space. It connects the full (or "dressed") electron propagator to the full electron-photon vertex.

Let's define our key objects:

• The full electron propagator, $S(p)$: This is the sum of all Feynman diagrams for an electron propagating from one point to another, including all self-interactions. It is the inverse of the 1PI two-point function, $S(p)=i(/p-m-\Sigma (p){)}^{-1}$, where $\Sigma (p)$ is the electron self-energy.
• The 1PI vertex function, ${\Gamma }^{\mu }(p^{\prime },p)$: This represents the full interaction between an electron and a photon, including all possible loop corrections. It is the "amputated" three-point function, meaning the external leg propagators have been removed.

By applying the general Ward-Takahashi identity to the three-point function $⟨\Omega |T\{\hat{\psi }(x)\hat{\bar{\psi }}(y){\hat{J}}^{\mu }(z)\}|\Omega ⟩$ and then Fourier transforming to momentum space, we arrive at a remarkably simple and powerful relation.

The following reading guides you through this specific derivation.

The result of this derivation is the momentum-space Ward-Takahashi identity:

where $p$ is the incoming electron momentum, $p^{\prime }$ is the outgoing electron momentum, and $q=p^{\prime }-p$ is the photon momentum. This equation relates a three-point function (${\Gamma }^{\mu }$) to two-point functions ($S^{-1}$). It is an exact, non-perturbative statement about the structure of QED.

A Profound Consequence: Renormalization and $Z_1=Z_2$

What is the physical significance of this identity? One of its most critical roles is in the theory of renormalization. In QED, we define several renormalization constants:

• $Z_1$: The vertex renormalization constant.
• $Z_2$: The electron field (or wave function) renormalization constant.
• $Z_3$: The photon field renormalization constant.

These constants are, a priori, independent. However, the Ward-Takahashi identity links the vertex correction to the propagator correction, which in turn imposes a powerful constraint: $Z_1=Z_2$.

This equality is not a minor technical detail; it is essential for the consistency of QED. It ensures that the renormalization of the electron's charge comes only from the photon propagator corrections ($Z_3$), a phenomenon known as vacuum polarization. The charge universality—the fact that all fundamental particles have charges that are integer multiples of a basic unit—is protected by this identity.

Let's walk through the proof of this crucial result.

The differential form of the Ward identity, ${\Gamma }^{\mu }(p,p)=\partial S^{-1}(p)/\partial p_{\mu }$, can be represented diagrammatically.

Conclusion

In this lesson, we have uncovered one of the deepest consequences of symmetry in quantum field theory. We saw that the classical conservation of charge, arising from a global U(1) symmetry, is elevated to a powerful set of constraints on the quantum theory's correlation functions, known as the Ward-Takahashi identities.

Key Takeaways:

• The Ward-Takahashi identity is the quantum statement of charge conservation, derived from the global U(1) symmetry of the QED Lagrangian.
• In coordinate space, it states that the divergence of a current correlation function vanishes except at the insertion points of other charged operators, where it produces "contact terms."
• This identity can be derived either by analyzing the action of derivatives on time-ordered products or by considering a position-dependent field redefinition in the path integral.
• In momentum space, it yields specific relations between Green's functions. The most famous is $-i{q}_{\mu }{\Gamma }^{\mu }(p^{\prime },p)=S^{-1}(p^{\prime })-S^{-1}(p)$, which relates the full electron-photon vertex ${\Gamma }^{\mu }$ to the full electron propagator $S$.
• A profound physical consequence of this identity is the relation $Z_1=Z_2$ between the vertex and field renormalization constants in QED, which is crucial for the consistency and predictive power of the theory.

You have now completed the journey from classical symmetries to their powerful manifestations in quantum gauge theories. You've seen how demanding a local symmetry builds the theory of QED, and how the remaining global symmetry constrains it.

Preview of the Next Module:

Our exploration has so far centered on the simplest continuous symmetry group, U(1). In the next module, "Gauge Theories and Fundamental Symmetries," we will generalize this entire framework. We will replace U(1) with non-abelian Lie groups like SU(N), leading us to the discovery of Yang-Mills theories. This is the final step needed to construct the theoretical framework for the weak and strong nuclear forces, completing the foundation of the Standard Model of particle physics.