QFT Bridge: EFT and Effective Hamiltonians

Effective Hamiltonians and Scale Separation develops the quantum-mechanical method map. Effective Hamiltonians prepare the core logic of effective field theory: keep the degrees of freedom relevant at a chosen scale, encode eliminated physics in effective operators, and determine coefficients by matching observables.

The analogy is conceptual rather than literal. A quantum-mechanical effective Hamiltonian acts on a Hilbert-space subspace. A QFT effective field theory is usually a Lagrangian or Hamiltonian density with all operators allowed by symmetries, organized by a power counting.

Scale Separation in Quantum Mechanics

In a Schrieffer-Wolff construction, a Hilbert space is split into retained and eliminated sectors:

$$P+Q=I.$$

Virtual excursions into $Q$ generate corrections to the Hamiltonian acting in $P$. A typical second-order term has the structure

$$\frac{V_{PQ}V_{QP}}{\Delta},$$

where $\Delta$ is a large energy denominator.

The low-energy theory does not keep the high-energy states explicitly, but it remembers them through shifted energies and induced couplings.

Integrating Out

The Feshbach formalism gives the exact energy-dependent expression

$$H_{\mathrm{eff}}(E) = H_{PP} + H_{PQ} \frac{1}{E-H_{QQ}} H_{QP}.$$

Expanding the resolvent in powers of a small ratio produces local-looking corrections in the retained subspace. This is the quantum-mechanical prototype of integrating out heavy fields or high-energy modes in EFT.

In QFT, integrating out a heavy scale $M$ produces operators suppressed by powers of $1/M$:

$$\mathcal L_{\mathrm{eff}} = \mathcal L_{\mathrm{light}} + \sum_i \frac{c_i}{M^{d_i-4}} \mathcal O_i.$$

The coefficients $c_i$ encode the eliminated short-distance physics.

Matching

An effective description is not fixed by symmetry alone. Its coefficients must be matched to observables or to a more microscopic theory.

In low-energy scattering, the effective-range expansion matches threshold observables:

$$k\cot\delta_0(k) = - \frac1a + \frac12r_ek^2 + \cdots.$$

An EFT description of the same system chooses contact operators whose coefficients reproduce $a$, $r_e$, and higher shape parameters to the desired order.

Matching can be done to experiment, to an exact solution, to a numerical calculation, or to a more fundamental theory.

Power Counting

Power counting is the rule that says which operators are needed at a given accuracy. In quantum mechanics, the analogous question is: how many orders in $V/\Delta$, $kR$, or $1/\Omega$ must be retained?

Examples:

• Schrieffer-Wolff: organize terms by powers of coupling over gap.
• Effective-range expansion: organize terms by powers of $kR$.
• Floquet-Magnus: organize terms by powers of $1/\Omega$.
• WKB: organize terms by powers of $\hbar/S$.

EFT makes this bookkeeping systematic for fields.

What Changes in QFT

QFT adds features not present in elementary effective Hamiltonians:

• local operator bases constrained by spacetime symmetries;
• renormalization-scale dependence;
• loop corrections and counterterms;
• field redefinitions and redundant operators;
• matching across thresholds;
• many-particle states and particle production.

The effective-Hamiltonian intuition remains useful, but QFT requires a more precise treatment of locality and renormalization.

Common Mistakes

• Thinking “effective” means approximate in an uncontrolled way.
• Keeping a high-energy state explicitly and also including its induced operator, thereby double counting.
• Matching only one observable and assuming all others are fixed.
• Ignoring symmetry constraints on effective operators.
• Forgetting that EFT coefficients can depend on the renormalization scale while observables do not.

Exercises

1. What is the EFT analogue of a Schrieffer-Wolff energy denominator?

Solution

The analogue is suppression by a heavy scale or large gap. In Schrieffer-Wolff, virtual high-energy states generate terms proportional to powers of $1/\Delta$. In EFT, heavy fields or short-distance modes generate operators suppressed by powers of $1/M$, where $M$ is the scale that has been integrated out.

2. Why is matching necessary even after writing all symmetry-allowed operators?

Solution

Symmetry determines which operators may appear, but not their coefficients. The coefficients contain information about the short-distance physics or experimental input. Matching fixes those coefficients by requiring the effective theory to reproduce selected low-energy observables or amplitudes.

References

• S. Weinberg, The Quantum Theory of Fields, Vol. I: Foundations, Cambridge University Press, 1995.
• H. Georgi, “Effective field theory,” Annual Review of Nuclear and Particle Science 43, 209-252, 1993.
• A. V. Manohar, “Effective field theories,” Les Houches Lectures, arXiv:hep-ph/9606222, 1996.
• C. P. Burgess, Introduction to Effective Field Theory, Cambridge University Press, 2020.
• S. Bravyi, D. P. DiVincenzo, and D. Loss, “Schrieffer-Wolff transformation for quantum many-body systems,” Annals of Physics 326, 2793-2826, 2011.