Arai's Effective Hamiltonian in Non-Relativistic QED

• Arai’s effective Hamiltonian is a reduced operator in non-relativistic QED that replaces the external potential with a Gaussian average determined by the electron–field coupling.
• It utilizes projection techniques and characteristic polynomials to eliminate complementary degrees of freedom and improve perturbative analyticity.
• The formulation clarifies the interplay between dressed states, spectral invariants, and operator theory, addressing common misunderstandings in effective Hamiltonian approaches.

Searching arXiv for the cited papers and closely related formulations. arxiv_search.search(query="Arai effective Hamiltonian non-relativistic quantum electrodynamics (Matsuzawa, 24 Apr 2026, Zheng, 2022)", max_results=5) arxiv_search.search(query="Arai effective Hamiltonian non-relativistic quantum electrodynamics", max_results=10) Arai’s effective Hamiltonian denotes, in the most specific sense represented in recent non-relativistic quantum electrodynamics, a Schrödinger operator for a dressed electron in which the external potential is replaced by a Gaussian average determined by the electron–field coupling and ultraviolet cutoff. In a broader effective-Hamiltonian theory, the same name is associated with projection-based elimination of complementary degrees of freedom, typically through a $P/Q$ decomposition or a dressed-state construction. Within that broader setting, recent analysis emphasizes not merely a particular reduced operator but the characteristic polynomial of the effective Hamiltonian, whose symmetric structure removes certain singularities present in individual eigenvalue expansions and thereby improves perturbative analyticity (Matsuzawa, 24 Apr 2026, Zheng, 2022).

1. Definition and physical setting

In the non-relativistic QED formulation, the starting point is a simplified Pauli–Fierz Hamiltonian for a non-relativistic electron coupled to the quantized electromagnetic field. The total Hilbert space is

$$\mathscr H_{\mathrm{tot}}=L^2(\mathbb R^d_{\boldsymbol x})\otimes \mathscr F_{\mathrm b}(\mathscr H_{\mathrm{ph}}), \qquad \mathscr H_{\mathrm{ph}}=L^2(\mathbb R^d_{\boldsymbol k}\times\{1,\dots,d-1\}),$$

with the electron in $L^2(\mathbb R^d)$ and the photons in bosonic Fock space. The model uses the dipole approximation, omits the $\boldsymbol A^2$ term, and incorporates mass renormalization into the free part (Matsuzawa, 24 Apr 2026).

The total Hamiltonian is written as

$H_{\mathrm{tot}}=H_0\dot{+}(V\otimes 1),$

where

$$H_0= -\frac{\hbar^2}{2m_0}\Delta\otimes 1 +1\otimes \mathrm d\Gamma_{\mathrm b}(\hbar\omega) -\frac{q}{m}\sum_{j=1}^d p_j\otimes A_j(\boldsymbol 0).$$

Here $p_j=-i\hbar\partial_{x_j}$, $\omega(\boldsymbol k)>0$ is the photon dispersion, and $A_j(\boldsymbol 0)=\Phi_{\mathrm S}(g_j)$ is the Segal field at the origin. The coupling functions are

$$g_j(\boldsymbol k,\lambda) = \sqrt{\frac{\hbar}{\epsilon_0\,\omega(\boldsymbol k)}}\,\hat\rho(\boldsymbol k)\, e_j^{(\lambda)}(\boldsymbol k),$$

with the assumptions

$$\mathscr H_{\mathrm{tot}}=L^2(\mathbb R^d_{\boldsymbol x})\otimes \mathscr F_{\mathrm b}(\mathscr H_{\mathrm{ph}}), \qquad \mathscr H_{\mathrm{ph}}=L^2(\mathbb R^d_{\boldsymbol k}\times\{1,\dots,d-1\}),$$0

The observed mass $$\mathscr H_{\mathrm{tot}}=L^2(\mathbb R^d_{\boldsymbol x})\otimes \mathscr F_{\mathrm b}(\mathscr H_{\mathrm{ph}}), \qquad \mathscr H_{\mathrm{ph}}=L^2(\mathbb R^d_{\boldsymbol k}\times\{1,\dots,d-1\}),$$1 and bare mass $$\mathscr H_{\mathrm{tot}}=L^2(\mathbb R^d_{\boldsymbol x})\otimes \mathscr F_{\mathrm b}(\mathscr H_{\mathrm{ph}}), \qquad \mathscr H_{\mathrm{ph}}=L^2(\mathbb R^d_{\boldsymbol k}\times\{1,\dots,d-1\}),$$2 are related by

$$\mathscr H_{\mathrm{tot}}=L^2(\mathbb R^d_{\boldsymbol x})\otimes \mathscr F_{\mathrm b}(\mathscr H_{\mathrm{ph}}), \qquad \mathscr H_{\mathrm{ph}}=L^2(\mathbb R^d_{\boldsymbol k}\times\{1,\dots,d-1\}),$$3

This choice is made so that the free dressed system has the effective kinetic energy $$\mathscr H_{\mathrm{tot}}=L^2(\mathbb R^d_{\boldsymbol x})\otimes \mathscr F_{\mathrm b}(\mathscr H_{\mathrm{ph}}), \qquad \mathscr H_{\mathrm{ph}}=L^2(\mathbb R^d_{\boldsymbol k}\times\{1,\dots,d-1\}),$$4. In this formulation, Arai’s effective Hamiltonian is the reduced electron Hamiltonian obtained after the photon field is encoded through dressing and Gaussian smoothing of the external potential (Matsuzawa, 24 Apr 2026).

2. Dressed-electron construction

The central mechanism is the dressed-electron state. After Fourier transform in the electron variable, the free Hamiltonian decomposes as

$$\mathscr H_{\mathrm{tot}}=L^2(\mathbb R^d_{\boldsymbol x})\otimes \mathscr F_{\mathrm b}(\mathscr H_{\mathrm{ph}}), \qquad \mathscr H_{\mathrm{ph}}=L^2(\mathbb R^d_{\boldsymbol k}\times\{1,\dots,d-1\}),$$5

with fiber Hamiltonians

$$\mathscr H_{\mathrm{tot}}=L^2(\mathbb R^d_{\boldsymbol x})\otimes \mathscr F_{\mathrm b}(\mathscr H_{\mathrm{ph}}), \qquad \mathscr H_{\mathrm{ph}}=L^2(\mathbb R^d_{\boldsymbol k}\times\{1,\dots,d-1\}),$$6

These are van Hove Hamiltonians and are diagonalized exactly by the Weyl-type unitary

$$\mathscr H_{\mathrm{tot}}=L^2(\mathbb R^d_{\boldsymbol x})\otimes \mathscr F_{\mathrm b}(\mathscr H_{\mathrm{ph}}), \qquad \mathscr H_{\mathrm{ph}}=L^2(\mathbb R^d_{\boldsymbol k}\times\{1,\dots,d-1\}),$$7

so that

$$\mathscr H_{\mathrm{tot}}=L^2(\mathbb R^d_{\boldsymbol x})\otimes \mathscr F_{\mathrm b}(\mathscr H_{\mathrm{ph}}), \qquad \mathscr H_{\mathrm{ph}}=L^2(\mathbb R^d_{\boldsymbol k}\times\{1,\dots,d-1\}),$$8

The fiber ground state is unique,

$$\mathscr H_{\mathrm{tot}}=L^2(\mathbb R^d_{\boldsymbol x})\otimes \mathscr F_{\mathrm b}(\mathscr H_{\mathrm{ph}}), \qquad \mathscr H_{\mathrm{ph}}=L^2(\mathbb R^d_{\boldsymbol k}\times\{1,\dots,d-1\}),$$9

with $L^2(\mathbb R^d)$0 the Fock vacuum. Its ground energy is

$L^2(\mathbb R^d)$1

and, using the mass relation above,

$L^2(\mathbb R^d)$2

Hence $L^2(\mathbb R^d)$3 is unitarily equivalent to

$L^2(\mathbb R^d)$4

Given $L^2(\mathbb R^d)$5, the dressed state is

$L^2(\mathbb R^d)$6

Equivalently, in position space,

$L^2(\mathbb R^d)$7

This construction is the operative form of Arai’s original idea: each electron momentum component is dressed by the photon-fiber ground state carrying that same momentum. A plausible implication is that the reduced dynamics is not obtained by a purely formal elimination of photons but by an explicit embedding of electron states into the coupled electron–photon Hilbert space.

3. Gaussian effective potential and operator-theoretic properties

The effective potential is defined by Gaussian averaging: $L^2(\mathbb R^d)$8 with

$L^2(\mathbb R^d)$9

Thus $\boldsymbol A^2$0 is exactly the convolution of $\boldsymbol A^2$1 with a Gaussian heat kernel, and the corresponding effective Hamiltonian is

$\boldsymbol A^2$2

The quadratic form sum exists; accordingly, $\boldsymbol A^2$3 is self-adjoint and bounded below (Matsuzawa, 24 Apr 2026).

The derivation proceeds directly from quadratic forms on dressed states. If $\boldsymbol A^2$4 is the quadratic form of $\boldsymbol A^2$5, one defines

$\boldsymbol A^2$6

Two identities are decisive. For the kinetic term,

$\boldsymbol A^2$7

For the potential term,

$\boldsymbol A^2$8

The latter is based on the overlap formula

$\boldsymbol A^2$9

whose Gaussian factor is the Fourier transform of the heat kernel.

The admissible external potentials are broad. The assumptions are either that $H_{\mathrm{tot}}=H_0\dot{+}(V\otimes 1),$0 is infinitesimally form-bounded with respect to $H_{\mathrm{tot}}=H_0\dot{+}(V\otimes 1),$1, or that $H_{\mathrm{tot}}=H_0\dot{+}(V\otimes 1),$2 is bounded from below, together with

$H_{\mathrm{tot}}=H_0\dot{+}(V\otimes 1),$3

This includes the Rollnik class in $H_{\mathrm{tot}}=H_0\dot{+}(V\otimes 1),$4, hence $H_{\mathrm{tot}}=H_0\dot{+}(V\otimes 1),$5, and the harmonic oscillator potential

$H_{\mathrm{tot}}=H_0\dot{+}(V\otimes 1),$6

For the harmonic oscillator,

$H_{\mathrm{tot}}=H_0\dot{+}(V\otimes 1),$7

so the field contributes only a constant energy shift. The same work also gives the spectral comparison

$H_{\mathrm{tot}}=H_0\dot{+}(V\otimes 1),$8

4. Projection-space formulation and characteristic polynomial

A more general effective-Hamiltonian theory begins from

$H_{\mathrm{tot}}=H_0\dot{+}(V\otimes 1),$9

together with a decomposition of the model space into a $$H_0= -\frac{\hbar^2}{2m_0}\Delta\otimes 1 +1\otimes \mathrm d\Gamma_{\mathrm b}(\hbar\omega) -\frac{q}{m}\sum_{j=1}^d p_j\otimes A_j(\boldsymbol 0).$$0-space of dimension $$H_0= -\frac{\hbar^2}{2m_0}\Delta\otimes 1 +1\otimes \mathrm d\Gamma_{\mathrm b}(\hbar\omega) -\frac{q}{m}\sum_{j=1}^d p_j\otimes A_j(\boldsymbol 0).$$1 and a complementary $$H_0= -\frac{\hbar^2}{2m_0}\Delta\otimes 1 +1\otimes \mathrm d\Gamma_{\mathrm b}(\hbar\omega) -\frac{q}{m}\sum_{j=1}^d p_j\otimes A_j(\boldsymbol 0).$$2-space of dimension $$H_0= -\frac{\hbar^2}{2m_0}\Delta\otimes 1 +1\otimes \mathrm d\Gamma_{\mathrm b}(\hbar\omega) -\frac{q}{m}\sum_{j=1}^d p_j\otimes A_j(\boldsymbol 0).$$3. The objective is to eliminate the $$H_0= -\frac{\hbar^2}{2m_0}\Delta\otimes 1 +1\otimes \mathrm d\Gamma_{\mathrm b}(\hbar\omega) -\frac{q}{m}\sum_{j=1}^d p_j\otimes A_j(\boldsymbol 0).$$4-space and obtain an operator acting only in the $$H_0= -\frac{\hbar^2}{2m_0}\Delta\otimes 1 +1\otimes \mathrm d\Gamma_{\mathrm b}(\hbar\omega) -\frac{q}{m}\sum_{j=1}^d p_j\otimes A_j(\boldsymbol 0).$$5-space. In the wave-operator construction,

$$H_0= -\frac{\hbar^2}{2m_0}\Delta\otimes 1 +1\otimes \mathrm d\Gamma_{\mathrm b}(\hbar\omega) -\frac{q}{m}\sum_{j=1}^d p_j\otimes A_j(\boldsymbol 0).$$6

and the effective Hamiltonian is

$$H_0= -\frac{\hbar^2}{2m_0}\Delta\otimes 1 +1\otimes \mathrm d\Gamma_{\mathrm b}(\hbar\omega) -\frac{q}{m}\sum_{j=1}^d p_j\otimes A_j(\boldsymbol 0).$$7

The wave operator satisfies

$$H_0= -\frac{\hbar^2}{2m_0}\Delta\otimes 1 +1\otimes \mathrm d\Gamma_{\mathrm b}(\hbar\omega) -\frac{q}{m}\sum_{j=1}^d p_j\otimes A_j(\boldsymbol 0).$$8

and the perturbative components obey

$$H_0= -\frac{\hbar^2}{2m_0}\Delta\otimes 1 +1\otimes \mathrm d\Gamma_{\mathrm b}(\hbar\omega) -\frac{q}{m}\sum_{j=1}^d p_j\otimes A_j(\boldsymbol 0).$$9

The recent emphasis is that many effective Hamiltonians are possible, related by similarity transformations and differing in whether they are energy-dependent or independent, Hermitian or non-Hermitian; the characteristic polynomial is therefore treated as the central invariant object rather than any single matrix representation (Zheng, 2022).

That characteristic polynomial is

$p_j=-i\hbar\partial_{x_j}$0

where the coefficients $p_j=-i\hbar\partial_{x_j}$1 are the symmetric polynomials of the $p_j=-i\hbar\partial_{x_j}$2-space eigenvalues. For example,

$p_j=-i\hbar\partial_{x_j}$3

The central analytic claim is that the individual $p_j=-i\hbar\partial_{x_j}$4-space eigenvalues have branch points caused by $p_j=-i\hbar\partial_{x_j}$5-crossings and $p_j=-i\hbar\partial_{x_j}$6-crossings, whereas the coefficients $p_j=-i\hbar\partial_{x_j}$7 and $p_j=-i\hbar\partial_{x_j}$8, being symmetric under permutation of $p_j=-i\hbar\partial_{x_j}$9-space eigenvalues, do not acquire branch points from $\omega(\boldsymbol k)>0$0-crossings. Their only branch points are those caused by $\omega(\boldsymbol k)>0$1-crossings. Consequently, if

$\omega(\boldsymbol k)>0$2

is the common convergence radius for the eigenvalue series, then for the polynomial coefficients

$\omega(\boldsymbol k)>0$3

with

$\omega(\boldsymbol k)>0$4

The paper explicitly notes that the inequality is often strict, even $\omega(\boldsymbol k)>0$5.

This formalism also addresses the intruder-state problem. In perturbative expansions of individual $\omega(\boldsymbol k)>0$6-space eigenvalues, denominators such as

$\omega(\boldsymbol k)>0$7

appear. These are the familiar intruder-state terms. When the symmetric combinations $\omega(\boldsymbol k)>0$8 are formed, those terms cancel. For $\omega(\boldsymbol k)>0$9,

$A_j(\boldsymbol 0)=\Phi_{\mathrm S}(g_j)$0

and

$A_j(\boldsymbol 0)=\Phi_{\mathrm S}(g_j)$1

The relevant coalescence condition is then

$A_j(\boldsymbol 0)=\Phi_{\mathrm S}(g_j)$2

An explicit effective Hamiltonian sharing exactly the same singularities as the characteristic polynomial is given by the companion-matrix form

$A_j(\boldsymbol 0)=\Phi_{\mathrm S}(g_j)$3

whose characteristic polynomial is the same as that of $A_j(\boldsymbol 0)=\Phi_{\mathrm S}(g_j)$4. This construction makes explicit that one can choose an effective Hamiltonian with the same branch points and the same convergence radius $A_j(\boldsymbol 0)=\Phi_{\mathrm S}(g_j)$5 as the polynomial itself.

5. Time-dependent projected Hamiltonians

A related projected-dynamics formulation appears in the treatment of unstable subsystems. Here one begins from the full Schrödinger evolution

$A_j(\boldsymbol 0)=\Phi_{\mathrm S}(g_j)$6

with a projection $A_j(\boldsymbol 0)=\Phi_{\mathrm S}(g_j)$7 onto the unstable subspace $A_j(\boldsymbol 0)=\Phi_{\mathrm S}(g_j)$8 and $A_j(\boldsymbol 0)=\Phi_{\mathrm S}(g_j)$9 onto the complement. The projected evolution operator is

$$g_j(\boldsymbol k,\lambda) = \sqrt{\frac{\hbar}{\epsilon_0\,\omega(\boldsymbol k)}}\,\hat\rho(\boldsymbol k)\, e_j^{(\lambda)}(\boldsymbol k),$$0

Projecting the full Schrödinger equation yields the Krolikowski–Rzewuski equation

$$g_j(\boldsymbol k,\lambda) = \sqrt{\frac{\hbar}{\epsilon_0\,\omega(\boldsymbol k)}}\,\hat\rho(\boldsymbol k)\, e_j^{(\lambda)}(\boldsymbol k),$$1

with memory kernel

$$g_j(\boldsymbol k,\lambda) = \sqrt{\frac{\hbar}{\epsilon_0\,\omega(\boldsymbol k)}}\,\hat\rho(\boldsymbol k)\, e_j^{(\lambda)}(\boldsymbol k),$$2

The differential form is

$$g_j(\boldsymbol k,\lambda) = \sqrt{\frac{\hbar}{\epsilon_0\,\omega(\boldsymbol k)}}\,\hat\rho(\boldsymbol k)\, e_j^{(\lambda)}(\boldsymbol k),$$3

so the effective Hamiltonian is in general time-dependent and non-Hermitian (Urbanowski, 2014).

Under the leading approximation $$g_j(\boldsymbol k,\lambda) = \sqrt{\frac{\hbar}{\epsilon_0\,\omega(\boldsymbol k)}}\,\hat\rho(\boldsymbol k)\, e_j^{(\lambda)}(\boldsymbol k),$$4,

$$g_j(\boldsymbol k,\lambda) = \sqrt{\frac{\hbar}{\epsilon_0\,\omega(\boldsymbol k)}}\,\hat\rho(\boldsymbol k)\, e_j^{(\lambda)}(\boldsymbol k),$$5

For an $$g_j(\boldsymbol k,\lambda) = \sqrt{\frac{\hbar}{\epsilon_0\,\omega(\boldsymbol k)}}\,\hat\rho(\boldsymbol k)\, e_j^{(\lambda)}(\boldsymbol k),$$6-dimensional subspace with

$$g_j(\boldsymbol k,\lambda) = \sqrt{\frac{\hbar}{\epsilon_0\,\omega(\boldsymbol k)}}\,\hat\rho(\boldsymbol k)\, e_j^{(\lambda)}(\boldsymbol k),$$7

the large-time limit gives

$$g_j(\boldsymbol k,\lambda) = \sqrt{\frac{\hbar}{\epsilon_0\,\omega(\boldsymbol k)}}\,\hat\rho(\boldsymbol k)\, e_j^{(\lambda)}(\boldsymbol k),$$8

hence

$$g_j(\boldsymbol k,\lambda) = \sqrt{\frac{\hbar}{\epsilon_0\,\omega(\boldsymbol k)}}\,\hat\rho(\boldsymbol k)\, e_j^{(\lambda)}(\boldsymbol k),$$9

If $$\mathscr H_{\mathrm{tot}}=L^2(\mathbb R^d_{\boldsymbol x})\otimes \mathscr F_{\mathrm b}(\mathscr H_{\mathrm{ph}}), \qquad \mathscr H_{\mathrm{ph}}=L^2(\mathbb R^d_{\boldsymbol k}\times\{1,\dots,d-1\}),$$00 is $$\mathscr H_{\mathrm{tot}}=L^2(\mathbb R^d_{\boldsymbol x})\otimes \mathscr F_{\mathrm b}(\mathscr H_{\mathrm{ph}}), \qquad \mathscr H_{\mathrm{ph}}=L^2(\mathbb R^d_{\boldsymbol k}\times\{1,\dots,d-1\}),$$01-fold degenerate with eigenvalue $$\mathscr H_{\mathrm{tot}}=L^2(\mathbb R^d_{\boldsymbol x})\otimes \mathscr F_{\mathrm b}(\mathscr H_{\mathrm{ph}}), \qquad \mathscr H_{\mathrm{ph}}=L^2(\mathbb R^d_{\boldsymbol k}\times\{1,\dots,d-1\}),$$02, then

$$\mathscr H_{\mathrm{tot}}=L^2(\mathbb R^d_{\boldsymbol x})\otimes \mathscr F_{\mathrm b}(\mathscr H_{\mathrm{ph}}), \qquad \mathscr H_{\mathrm{ph}}=L^2(\mathbb R^d_{\boldsymbol k}\times\{1,\dots,d-1\}),$$03

The one-dimensional case yields the exact formula

$$\mathscr H_{\mathrm{tot}}=L^2(\mathbb R^d_{\boldsymbol x})\otimes \mathscr F_{\mathrm b}(\mathscr H_{\mathrm{ph}}), \qquad \mathscr H_{\mathrm{ph}}=L^2(\mathbb R^d_{\boldsymbol k}\times\{1,\dots,d-1\}),$$04

which is the exact one-particle effective Hamiltonian. Its real part is the instantaneous energy,

$$\mathscr H_{\mathrm{tot}}=L^2(\mathbb R^d_{\boldsymbol x})\otimes \mathscr F_{\mathrm b}(\mathscr H_{\mathrm{ph}}), \qquad \mathscr H_{\mathrm{ph}}=L^2(\mathbb R^d_{\boldsymbol k}\times\{1,\dots,d-1\}),$$05

and its imaginary part determines the instantaneous decay width,

$$\mathscr H_{\mathrm{tot}}=L^2(\mathbb R^d_{\boldsymbol x})\otimes \mathscr F_{\mathrm b}(\mathscr H_{\mathrm{ph}}), \qquad \mathscr H_{\mathrm{ph}}=L^2(\mathbb R^d_{\boldsymbol k}\times\{1,\dots,d-1\}),$$06

When $$\mathscr H_{\mathrm{tot}}=L^2(\mathbb R^d_{\boldsymbol x})\otimes \mathscr F_{\mathrm b}(\mathscr H_{\mathrm{ph}}), \qquad \mathscr H_{\mathrm{ph}}=L^2(\mathbb R^d_{\boldsymbol k}\times\{1,\dots,d-1\}),$$07, the asymptotic expansion

$$\mathscr H_{\mathrm{tot}}=L^2(\mathbb R^d_{\boldsymbol x})\otimes \mathscr F_{\mathrm b}(\mathscr H_{\mathrm{ph}}), \qquad \mathscr H_{\mathrm{ph}}=L^2(\mathbb R^d_{\boldsymbol k}\times\{1,\dots,d-1\}),$$08

implies

$$\mathscr H_{\mathrm{tot}}=L^2(\mathbb R^d_{\boldsymbol x})\otimes \mathscr F_{\mathrm b}(\mathscr H_{\mathrm{ph}}), \qquad \mathscr H_{\mathrm{ph}}=L^2(\mathbb R^d_{\boldsymbol k}\times\{1,\dots,d-1\}),$$09

This distinguishes the exact projected Hamiltonian from constant Weisskopf–Wigner or LOY effective Hamiltonians, which are accurate only in the exponential regime.

6. Interpretation, scope, and common misunderstandings

The literature represented here supports two complementary understandings of Arai’s effective-Hamiltonian program. In the non-relativistic QED setting, the name refers to the explicit operator

$$\mathscr H_{\mathrm{tot}}=L^2(\mathbb R^d_{\boldsymbol x})\otimes \mathscr F_{\mathrm b}(\mathscr H_{\mathrm{ph}}), \qquad \mathscr H_{\mathrm{ph}}=L^2(\mathbb R^d_{\boldsymbol k}\times\{1,\dots,d-1\}),$$10

derived by dressing electron states with exact fiber ground states and evaluating the full Hamiltonian on those dressed states. In the more abstract projection setting, the effective Hamiltonian is an operator acting in a chosen model space, typically obtained by wave operators, projected evolution equations, or similarity transformations, with recent work stressing that the characteristic polynomial may be the analytically most robust object (Matsuzawa, 24 Apr 2026, Zheng, 2022).

A frequent misunderstanding is to treat “the” effective Hamiltonian as unique. The projection-space analysis explicitly states that there are many possible effective Hamiltonians, related by similarity transformations, and that they may be energy-dependent or independent, Hermitian or non-Hermitian. This suggests that the invariant content lies less in any preferred matrix form than in spectral data, singularity structure, and the manner in which the eliminated sector is encoded.

A second misunderstanding is to identify the reduced operator with a mere low-order perturbative approximation. In the direct non-relativistic QED derivation, the effective Hamiltonian is obtained without using Arai’s scaling limit; instead, it follows from exact diagonalization of the free Pauli–Fierz part and exact quadratic-form identities on dressed states (Matsuzawa, 24 Apr 2026). Conversely, in unstable-state dynamics the effective Hamiltonian is intrinsically time-dependent, and constant effective Hamiltonians fail at very late times, where the exact projected Hamiltonian satisfies

$$\mathscr H_{\mathrm{tot}}=L^2(\mathbb R^d_{\boldsymbol x})\otimes \mathscr F_{\mathrm b}(\mathscr H_{\mathrm{ph}}), \qquad \mathscr H_{\mathrm{ph}}=L^2(\mathbb R^d_{\boldsymbol k}\times\{1,\dots,d-1\}),$$11

Taken together, these developments place Arai’s effective Hamiltonian within a general operator-theoretic strategy: one identifies a physically relevant subspace or dressed sector, integrates out the complementary degrees of freedom, and studies the resulting reduced dynamics through the effective operator or, when analyticity is decisive, through its characteristic polynomial. The recurring advantages are explicit decoupling, improved perturbative behavior, and a precise relation between the reduced description and the spectrum or dynamics of the full Hamiltonian.