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 $P$. If the eliminated states lie an energy $\Delta>0$ above a nearly degenerate low-energy cluster, the leading shift has the structure

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

The negative sign expresses level repulsion from higher states. General signed denominators must be retained when the eliminated spectrum is not a single high-energy cluster.

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

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

The inverse requires $E$ outside the eliminated spectrum; an open channel uses $E+i0$ and can produce an absorptive effective operator. Expanding a gapped resolvent in low energy divided by the gap gives a controlled low-energy series. A gap alone does not prove spatial locality; the range and analytic dependence of the microscopic interactions also matter. This is the quantum-mechanical prototype of integrating out heavy fields or high-energy modes in EFT.

In $3+1$ spacetime dimensions and units $\hbar=c=1$, a local low-energy expansion takes the form

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

Here $d_i$ is the operator mass dimension and the displayed $c_i$ are dimensionless. Operators with $d_i>4$ are suppressed by inverse powers of $M$; relevant and marginal terms can instead receive unsuppressed shifts and belong in the matched light theory. The local series applies below the heavy threshold and within its momentum expansion.

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 dimensionless drive scales divided by $\hbar\Omega$, within the specified time range.
• WKB: organize terms by $\hbar$ divided by a characteristic action, with separate turning-point matching.

EFT applies such scaling rules to operators and amplitudes. A large scattering length is an important exception to naive weak-coupling counting: the leading contact interaction may need iteration even when $kR\ll1$. Higher operators can still be treated systematically. Coefficients depend on the regulator and renormalization scale; matched observables do not.

What Changes in QFT

QFT makes the following features central; related structures can also occur in many-body quantum mechanics:

• 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.

Exercises

1. A two-level Hamiltonian has diagonal entries $0,\Delta$ with $\Delta>0$ and off-diagonal entries $v,v^*$. What low-energy shift follows by eliminating the upper state, and why is it negative?

Solution

The exact projected equation is $E=|v|^2/(E-\Delta)$. For $|E|\ll\Delta$, it gives $E\approx-|v|^2/\Delta$. The next term in the exact low eigenvalue is $+|v|^4/\Delta^3$. The high state is absent from the reduced space, but its virtual coupling lowers the retained energy.

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. Bravyi, D. P. DiVincenzo, and D. Loss, “Schrieffer–Wolff transformation for quantum many-body systems,” Annals of Physics 326, 2793-2826, 2011.
• C. P. Burgess, Introduction to Effective Field Theory, Cambridge University Press, 2020.
• H. Georgi, “Effective field theory,” Annual Review of Nuclear and Particle Science 43, 209-252, 1993.
• A. V. Manohar, “Effective field theories,” Schladming Winter School lectures, report UCSD/PTH 96-04 (1996). arXiv:hep-ph/9606222.
• S. Weinberg, The Quantum Theory of Fields, Vol. I: Foundations, Cambridge University Press, 1995.