Effective Hamiltonians

Eliminating the lower Dirac components produces an exact energy-dependent stationary equation when the required inverse exists. It also changes the norm with which the retained components represent a physical state. The reduction alone does not select positive energies, and it is not automatically a Hamiltonian for arbitrary time evolution. The generic Schur-complement derivation belongs to Projection Methods; this page applies that construction to the ordered Dirac blocks and their nonrelativistic energy window.

Required background. Projection Methods provides elimination and reconstruction; Dirac to Pauli fixes the upper/lower equations and signed electromagnetic coupling.

Helpful background. Nonrelativistic Limit specifies the comparison being made; Foldy–Wouthuysen Transformation and Schrieffer–Wolff Transformation explain the unitary alternative.

The stationary Dirac Schur operator

Use a static electromagnetic background, the Dirac basis, and $M=mc^2>0$. Put

$$V=q\Phi,\qquad \boldsymbol\pi=-i\hbar\nabla-q\mathbf A,\qquad W=c\,\boldsymbol\sigma\cdot\boldsymbol\pi.$$

For a self-adjoint realization on appropriate domains, the Hamiltonian blocks are

$$H=\begin{pmatrix}M+V&W\\W&-M+V\end{pmatrix}.$$

Seek a stationary state $\Psi=(\varphi,\chi)^T$ with total energy $E=M+\varepsilon$. In an energy window where $D_\varepsilon=2M+\varepsilon-V$ has the needed inverse, the lower equation and the Schur operator give

$$\chi=D_\varepsilon^{-1}W\varphi,$$ $$h_{\rm eff}(\varepsilon)\varphi=\varepsilon\varphi, \qquad h_{\rm eff}(\varepsilon) =V+W D_\varepsilon^{-1}W.$$

The order of these factors is physical: $[\boldsymbol\sigma\cdot\boldsymbol\pi,V]$ generally does not vanish. One cannot move the position-dependent denominator through a momentum operator.

For bounded real $V$, a condition such as $\inf_{\mathbf x}|2M+\varepsilon-V(\mathbf x)|>0$ ensures a bounded multiplication inverse. The products involving the unbounded $W$ still require compatible domains. A finite-dimensional regulator makes the displayed algebra immediate, but estimates for that regulator need uniform control before a continuum claim.

The retained projector here selects upper components: $P_{\rm upper}=\operatorname{diag}(I_2,0)$. It is not the positive spectral projector of $H$. The reduced nonlinear eigenvalue equation can encode negative as well as positive roots whenever the inverse and reconstruction exist. The intended particle branch must be chosen by its energy window and physical preparation.

The physical norm of the retained components

The reconstruction map at a real admissible energy is

$$\Omega_\varepsilon\varphi =\begin{pmatrix} \varphi\\D_\varepsilon^{-1}W\varphi \end{pmatrix}.$$

Substitution into the norm formula from Projection Methods gives

$$\begin{aligned} \|\Psi\|^2&=\langle\varphi,G_\varepsilon\varphi\rangle,\\ G_\varepsilon &=I+W D_\varepsilon^{-2}W =I-\partial_\varepsilon h_{\rm eff}. \end{aligned}$$

The derivative sign is negative: $\partial_\varepsilon D_\varepsilon^{-1} =-D_\varepsilon^{-2}$. The upper component of a normalized full state therefore does not generally have unit ordinary norm.

An observable $A$ must also be reconstructed. At this energy its matrix elements use $\Omega_\varepsilon^\dagger A\Omega_\varepsilon$. If one introduces the ordinarily normalized retained variable $\widetilde\varphi=G_\varepsilon^{1/2}\varphi$, the corresponding operator is

$$A_{\rm eff} =G_\varepsilon^{-1/2} \Omega_\varepsilon^\dagger A\Omega_\varepsilon G_\varepsilon^{-1/2}.$$

These formulas are energy-specific. They do not yet define one energy-independent unitary representation for arbitrary superpositions of different eigenstates. A systematic Foldy–Wouthuysen construction addresses that different task.

An ordered finite-matrix check

A small matrix can expose errors that a commuting free example misses. In a chosen energy unit, take the regulated blocks

$$M=3,\qquad V=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\qquad W=\begin{pmatrix}0&3\\3&0\end{pmatrix}.$$

This is a block-algebra example, not a discretization accuracy claim or a small-velocity regime. The upper basis vector $e_1$ couples to the lower $e_2$ through the full block $\left(\begin{smallmatrix}4&3\\3&-4\end{smallmatrix}\right)$, whose energies are $E=\pm5$.

For $E=5$, the residual energy is $\varepsilon=2$ and $D_\varepsilon=\operatorname{diag}(7,9)$. Taking $\varphi=e_1$ gives

$$\chi=\frac13 e_2,\qquad h_{\rm eff}(2)e_1=2e_1,\qquad \|\Psi\|^2=\frac{10}{9}.$$

Thus the normalized full eigenvector is $(3,0,0,1)^T/\sqrt{10}$. By contrast, the wrongly reordered operator $V+W^2D_\varepsilon^{-1}$ gives $(1+9/7)e_1=(16/7)e_1$, which fails the eigenvalue equation.

For the negative root $E=-5$, $\varepsilon=-8$ and $D_\varepsilon=\operatorname{diag}(-3,-1)$. The same upper vector reconstructs $\chi=-3e_2$ and $h_{\rm eff}(-8)e_1=-8e_1$. Eliminating lower components has retained this negative root; it has not made the problem particle-only.

From an exact reduction to an expansion

Existence of $D_\varepsilon^{-1}$ is weaker than convergence of an expansion around $2M$. In a bounded setting, the geometric series

$$D_\varepsilon^{-1} =\frac{1}{2M} \sum_{n=0}^{\infty} \left[-\frac{\varepsilon-V}{2M}\right]^n$$

converges in operator norm if $\|\varepsilon-V\|/(2M)<1$. Its leading term gives $h_{\rm eff}\simeq V+W^2/(2M)$. The Dirac-to-Pauli derivation evaluates this ordered square and obtains the magnetic Pauli term. The subsequent normalized static corrections, including their magnetic power counting, belong to Foldy–Wouthuysen Expansion.

An energy-dependent stationary reduction should not be inserted into a dynamical equation by the unexamined replacement $\varepsilon\mapsto i\hbar\partial_t$. With time-dependent blocks, exact elimination includes the omitted sector’s initial state and a memory integral. Adiabatic Elimination owns that time-domain derivation. A local approximation requires its own ordering, preparation, and scale estimates.

Construction	What it provides	What still needs care
Stationary Schur reduction	Exact energy-dependent eigenvalue equation in an admissible window	Reconstruction, norm, domains, and selecting the intended roots
Unitary SW or FW reduction	A transformed Hamiltonian, states, and observables through a stated order	Gap, power counting, and the truncation error
NRQED matching	Low-energy field operators with coefficients fixed by amplitudes	Operator basis, retained degrees of freedom, and radiative or structure contributions

These are compatible tools with different outputs. The NRQED preview explains what field-theory matching can add beyond a minimally coupled classical Dirac Hamiltonian.

Exercises

Negative-root normalization. Normalize the negative eigenvector reconstructed in the finite-matrix example and find the probability in its upper component.

Solution

The unnormalized vector is $(1,0,0,-3)^T$, with norm squared ten. Its normalized form is $(1,0,0,-3)^T/\sqrt{10}$, and its upper-component probability is $1/10$. It still has a nonzero upper projection despite its negative total energy.

Inverse versus series. Let the bounded operator $\varepsilon-V$ have the single value $4M$. Does the exact denominator have an inverse? Does the displayed geometric expansion converge?

Solution

$D_\varepsilon=6M$ has inverse $1/(6M)$. The series ratio is $-2$, so the expansion around $2M$ does not converge. An exact stationary reduction can remain valid outside the proposed perturbative regime.

An observable check. In the positive-root example, evaluate the expectation of the projector onto the lower block using the retained upper variable.

Solution

For unnormalized $\varphi=e_1$, the reconstructed lower norm is $1/9$ and the full norm is $10/9$. Their ratio is $1/10$. Using only the upper component’s ordinary norm would incorrectly return $1/9$.

References

• Dusson, Geneviève, Israel Michael Sigal, and Benjamin Stamm. “The Feshbach-Schur Map and Perturbation Theory.” (2021). arXiv:2105.02058. Inverse hypotheses, isospectral reduction, and reconstruction.
• Foldy, Leslie L., and Siegfried A. Wouthuysen. “On the Dirac Theory of Spin 1/2 Particles and Its Non-Relativistic Limit.” Physical Review 78, 29–36 (1950). doi:10.1103/PhysRev.78.29. Unitary separation and transformed observables.
• Thaller, Bernd. The Dirac Equation. Springer (1992). doi:10.1007/978-3-662-02753-0. Spectral and domain framework for Dirac operators and their low-energy limits.