Reference and Bridge Pages

Use these pages when you need to find a formula, compare methods, check an error estimate, or translate a quantum-mechanical calculation into the language of quantum field theory. Each reference points back to a fuller explanation. Before using a compact formula, identify its observable, assumptions, and symbol conventions.

Helpful background. Approximation decision tree helps identify a suitable approach; small parameters and errors explains accuracy estimates; notation and conventions fixes the meanings of frequently reused symbols.

Find the right reference

What you need	Page
A result whose assumptions you already understand	Formula sheet
The meaning or dimensions of a symbol	Common symbols
Signed gaps, state normalization, WKB momenta, or resolvent signs	Notation and conventions
A comparison between plausible methods	Method comparison table
A starting point for an unfamiliar problem	Approximation decision tree
Help estimating what your calculation omitted	Error-estimate checklist

A decision tree suggests a method; it does not establish its accuracy. For example, weak mixing is controlled by a coupling divided by a relevant level separation, not by the coupling divided by an arbitrary absolute energy. Near degeneracy, keep the nearby levels together and diagonalize their unperturbed splittings plus their coupling.

Likewise, $O(\epsilon^2)$ describes scaling in a specified limit. Without a bound on its coefficient and a valid parameter range, it is not a numerical error bar at a chosen value of $\epsilon$.

Translate scattering conventions first

Before comparing amplitudes from different sources, consult the scattering convention dictionary. Check state normalization, momentum versus wave vector, incoming or outgoing boundary conditions, and the conversion from amplitude to flux. Two differently normalized intermediate amplitudes can predict the same cross section.

The S-matrix bridge then explains how in/out states and unitarity extend to relativistic field theory. Relativistic state normalization and processes with changing particle number require additional structure.

Four bridges to field theory

Born approximation and tree level compares a potential’s Fourier transform with a relativistic exchange amplitude. A shared denominator is a clue, not a complete identification: one must specify the underlying interaction and take its nonrelativistic limit with consistent external-state normalization.

Optical theorem and unitarity follows the same operator identity, $S^\dagger S=I$, into different amplitude conventions. The common principle survives, while phase space, channel sums, and normalization factors change.

EFT and effective Hamiltonians connects elimination of remote quantum states with matching low-energy field-theory operators. Locality, operator expansions, renormalization, and power counting add ingredients that are not supplied by a finite matrix projection alone.

Instantons and saddle points compares imaginary-time paths with stationary field configurations, including fluctuation modes and integration contours. Read the conceptual instanton bridge first if these field-theory features are new. Gauge freedom, topology, and renormalization make the field problem richer; not every Euclidean saddle has a topological interpretation.

For practice before crossing frameworks, use a worked approximation problem or a numerical comparison.

Exercises

A small matrix element. Two levels have unperturbed energies $100$ and $100.001$, and an off-diagonal perturbation of $0.002$, all in the same energy units. Which comparison matters, and what is a useful first step?

Solution

The coupling-to-gap ratio is $0.002/0.001=2$, so nondegenerate perturbation theory for this pair is uncontrolled despite the large absolute energies. Diagonalize the two-state block, including both diagonal energies and the off-diagonal coupling. Coupling to other levels still needs a separate check. A common shift of the energy origin changes neither this conclusion nor the mixing.

An order symbol. A result is $Q=Q_0[1+c\epsilon+O(\epsilon^2)]$ and $\epsilon=0.2$. Does it follow that the neglected correction is below four percent of $Q_0$?

Solution

No. The nominal next power is $\epsilon^2=0.04$, but its coefficient is unspecified and the asymptotic regime may not yet be reached. A numerical estimate requires more information: a remainder bound, higher terms under justified truncation, or a sufficiently accurate independent comparison.

References

• Bender, C. M., and S. A. Orszag. Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory. Springer (1999). Asymptotic notation and approximation methods.
• Peskin, M. E., and D. V. Schroeder. An Introduction to Quantum Field Theory. Addison-Wesley (1995). Relativistic amplitudes and their normalization.
• Sakurai, J. J., and J. Napolitano. Modern Quantum Mechanics. 3rd ed., Cambridge University Press (2020). Perturbation theory, transitions, and nonrelativistic scattering.
• Schwartz, M. D. Quantum Field Theory and the Standard Model. Cambridge University Press (2014). Scattering, effective field theory, and renormalization.