Definition
DeWitt notation is a condensed index formalism employed in theoretical physics, particularly in quantum field theory and the calculus of variations, to represent fields, functional derivatives, and related operations as if they were vectors and tensors in an abstract infinite‑dimensional space. The notation treats discrete labels and continuous spacetime arguments as a single composite index, allowing summation conventions to encompass both ordinary sums and integrals.
Overview
The formalism streamlines expressions involving functional derivatives, actions, and equations of motion. For a field φ with components φⁱ(x), the combined index i encodes both the field component type and the spacetime point x. A repeated index implies integration over the continuous variables as well as summation over discrete ones, e.g.,
$$ A^{i}B_{i} \equiv \sum_{\alpha}\int d^{d}x,A^{\alpha}(x)B_{\alpha}(x), $$
where α labels internal degrees of freedom. Within this framework the action functional, metric on field space, and gauge‑fixing terms can be written compactly, facilitating algebraic manipulations in the background‑field method, BRST quantization, and the Batalin‑Vilkovisky formalism.
Etymology / Origin
The notation is named after the American physicist Bryce DeWitt (1923–2004), who introduced it in the 1960s while developing functional methods for quantum gravity and gauge theories. It first appeared explicitly in DeWitt’s seminal paper “Dynamical Theory of Groups and Fields” (1965) and was later systematized in his monograph The Global Approach to Quantum Field Theory (2003). The convention reflects DeWitt’s emphasis on treating the space of field configurations as a geometric manifold.
Characteristics
| Feature | Description |
|---|---|
| Composite index | An index i represents both discrete labels (e.g., field components, group indices) and continuous spacetime points. |
| Summation convention | Repeated indices imply both summation over discrete labels and integration over continuous variables (often written with an implicit measure). |
| Functional derivatives | The derivative with respect to a field is denoted $\frac{\delta}{\delta\phi^{i}}$, mirroring ordinary partial derivatives in finite‑dimensional vector spaces. |
| Metric on field space | A symmetric object $G_{ij}$ (or its inverse $G^{ij}$) acts as a metric on the infinite‑dimensional configuration space, allowing inner products such as $G_{ij}\phi^{i}\phi^{j}$. |
| Compact action representation | Actions and variations can be expressed succinctly, e.g., $S[\phi]=\frac12 G_{ij}\phi^{i}\phi^{j}+V(\phi)$, with equations of motion $\frac{\delta S}{\delta\phi^{i}}=0$. |
| Applicability | Used in path‑integral formulations, background‑field expansions, gauge‑fixing, and the algebraic structure of BRST and BV quantizations. |
Related Topics
- Functional derivative and functional integral (path integral)
- Background field method
- BRST symmetry and gauge fixing
- Batalin‑Vilkovisky (BV) formalism
- Geometry of field space (e.g., field‑space connections, metrics)
- Effective action and renormalization group equations
- Quantum gravity and gauge theories (where DeWitt originally applied the notation)