Uniqueness of the time-smeared contravariant representation and its domain of validity in the renormalized Lee model

The paper 'Application of the Tamm-Dancoff-method to the Lee-model' (W. Heisenberg, Nuclear Physics 5, 1958) applied the 'new Tamm-Dancoff method' to the Lee model renormalized in the dipole-ghost case and found that the method reproduces the exact stationary states, while finite contravariant (bra-side) representations of those states in terms of renormalized operators exist only when the V-particle operator is smeared over a finite time interval of half-width $t_{0}$ with a weight $f(t)$ constrained only by an integral condition. Issues were left open: (i) the arbitrary choice of $f(t)$ gives a continuous family of contravariant representations and blocks a unique spatial matter density $\varphi^{*}(\mathbf{x})\sigma(\mathbf{x})$ for the stationary state; (ii) the domain of validity of the asymptotic relations behind the smearing, cutoff, and norm-cancellation formulas (the regime $t_{0}\ll 1/E_{0},\,1/m_{\theta}$) was not delimited. The problem remains open: whether a smearing weight selected by a criterion intrinsic to the renormalized theory can single out a unique contravariant representation and matter density, and what the precise validity domain of the asymptotic relations is.

Background

The Lee model, introduced by T. D. Lee in 1954, is a deliberately simplified quantum field theory in which a heavy fermion $V$ converts into a lighter fermion $N$ by emitting a neutral scalar meson $\theta$ ($V\rightleftharpoons N+\theta$). Because each fixed-particle-number sector involves only a finite chain of wavefunction components, the model is exactly solvable after mass and coupling renormalization, and it became a standard testbed for renormalization ideas. When the bare coupling exceeds a critical value, the V-particle renormalization constant diverges and the physical V state degenerates into a 'dipole ghost' — coalescing states of zero and negative norm, so the state space acquires an indefinite metric; the renormalized treatment of exactly this case was carried out by W. Heisenberg in 'Lee model and quantisation of non linear field equations' (1957).

The Tamm–Dancoff method, proposed around 1950, is a Fock-space truncation scheme: the state vector of an interacting field theory is expanded on a restricted set of particle-number components and the equations of motion are projected onto that subspace. Heisenberg's 'new Tamm-Dancoff method' (described in his 'Quantum Theory of Fields and Elementary Particles', 1957, and in Dürr, Heisenberg, Mitter, Schlieder and Yamazaki, 'Zur Theorie der Elementarteilchen', 1959) reformulated the program around time-ordered vacuum-to-state matrix elements ($\tau$-functions) of renormalized field operators, aiming at a fundamental nonlinear spinor equation. The obstacle there was that the spinor theory permits production of arbitrarily many quanta, forcing a further 'contraction' approximation of uncontrolled accuracy, whose eigenvalue error Heisenberg estimated only heuristically — of order 10–20% in the first approximation, by analogy with the anharmonic oscillator.

Heisenberg's 'Application of the Tamm-Dancoff-method to the Lee-model' (1958) used the dipole-ghost Lee model as an exactly solvable test case. Two findings matter here. First, in the Lee model at most one $\theta$ quantum is created or absorbed at each vertex, so the Tamm–Dancoff hierarchy closes without contractions, and the $\tau$-function equations coincide (up to explicit constants) with the Schrödinger-sector equations of the model: the method is exact there. Second, while the covariant representation of a stationary state $|\Phi\rangle$ by $\tau$-functions of renormalized operators is finite, the dual — 'contravariant' — representation, which expresses the same state as a bra-created vector acting on the vacuum, diverges if the renormalized V operator is inserted at a single time, because the Schrödinger coefficient $c$ diverges like $1/g_{0}$ in the renormalization limit ($g_{0}$ the bare coupling). A finite contravariant representation exists only if the renormalized V operator is smeared over a finite time interval,

$$|\Phi\rangle=\left(\int_{-t_{0}}^{t_{0}}\psi_{V_r}^{*}(t)\,f(t)\,\mathrm{d}t+\psi_{N}^{*}(0)\int\sigma(\mathbf{k})\,a^{*}(\mathbf{k},0)\,\mathrm{d}^{3}k\right)|0\rangle,$$

where $a^{*}(\mathbf{k},t)$ creates a $\theta$ meson, $\psi_{N}$ and $\psi_{V_r}$ are the N- and renormalized V-field operators, $f(t)$ is a weight supported in $|t|\le t_{0}$, and $\sigma(\mathbf{k})$ is a contravariant momentum amplitude (distinct from the Schrödinger amplitude $\varphi(\mathbf{k})$). The weight $f$ is constrained only by an integral condition arising from the renormalized commutation relations (Heisenberg (1958); $h^{+}$, and likewise $h^{-}$, is the boundary value of the spectral function defined in eq. (2) of that paper in terms of the function characteristic of the Lee model introduced in Heisenberg (1957)); in the narrow-smearing regime $t_{0}\ll 1/E_{0},\,1/m_{\theta}$ ($E_{0}$ a characteristic eigenvalue, $m_{\theta}$ the meson mass) the step-function choice $f(t)=f_{0}$ for $|t|\le t_{0}$ yields the area–width relation $f_{0}\,t_{0}=\frac{bg}{4}\ln\frac{C}{t_{0}m_{\theta}}$, with $g$ the renormalized coupling, $b$ a finite renormalized parameter, and $C$ a model constant. Time smearing acts as a low-pass filter, $\sigma(\mathbf{k})/\varphi(\mathbf{k})\to 1$ for $\omega\ll 1/t_{0}$ and $\to 0$ for $\omega\gg 1/t_{0}$ ($\omega$ the meson energy), and the state norm comes out finite and $t_{0}$-independent because an explicit logarithmic term cancels against the truncated high-frequency tail of $|\varphi|^{2}$ — all within the same asymptotic regime and using the high-frequency asymptotics of the spectral functions $h^{\pm}(\omega)$ (boundary values of the function whose zeros give the discrete eigenvalues) over the whole integration range. Because $f(t)$ is otherwise arbitrary, each stationary state admits a continuous family of contravariant representations, and no unique spatial 'matter density' $\varphi^{*}(\mathbf{x})\sigma(\mathbf{x})$ (the product of the covariant and contravariant position-space amplitudes) is thereby defined.

Sekine, 'Canonical and non-canonical method of quantisation in the Lee model' (1961), pursued the construction in the indefinite-metric Lee model: the equal-time anticommutator of the renormalized V field vanishes in the infinite-cutoff limit, and a Hamiltonian formalism built on the time-averaged V operator reproduces canonical (cut-off) dynamics for all vacuum-to-state matrix elements with $E\ll 1/\Delta t$. Sekine's averaging kernel is fixed, exactly like Heisenberg's $f(t)$, by a single integral condition plus a constant-window ansatz, and Sekine leaves the high-energy regime $E\gtrsim 1/\Delta t$ and the singularities near $t=0$ unresolved.

Problem Statement

The problem left open by Heisenberg (1958) — sharpened but not resolved by Sekine (1961) — is to put the time-smeared contravariant representation of the renormalized (dipole-ghost) Lee model on a unique and controlled footing. Concretely:

(1) Uniqueness. The smearing weight $f(t)$ is subject only to the integral condition imposed by the renormalized commutation relations, so a given stationary state admits a continuous family of contravariant representations, and correspondingly no unique spatial 'matter density' $\varphi^{*}(\mathbf{x})\sigma(\mathbf{x})$ is defined for the stationary state. Does a smearing weight selected by a criterion intrinsic to the renormalized theory — for instance its commutation relations, positivity of the state norm, or a symmetry — single out a unique contravariant representation, and hence a unique matter density, or is the non-uniqueness essential (no such criterion can exist within the theory)? A weight fixed by hand, such as the constant-window kernel of Sekine (1961), does not by itself count as such an intrinsic criterion.

(2) Domain of validity. The area–width relation $f_{0}\,t_{0}=\frac{bg}{4}\ln\frac{C}{t_{0}m_{\theta}}$, the low-pass behavior of $\sigma(\mathbf{k})/\varphi(\mathbf{k})$, and the $t_{0}$-independence of the norm were derived in the narrow-smearing regime $t_{0}\ll 1/E_{0},\,1/m_{\theta}$, using the high-frequency asymptotics of $h^{\pm}(\omega)$ over the whole integration range. Determine the precise domain of validity of these relations: their behavior for finite $t_{0}$ comparable to $1/E_{0}$ or $1/m_{\theta}$, and for matrix elements involving states with $E\gtrsim 1/t_{0}$, where they are expected to be singular near $t=0$; establish which of the 1958 formulas remain exact, which acquire controlled corrections, and which fail.

Both questions concern the same 1958 construction: removing the arbitrary choice of the smearing weight $f(t)$ and delimiting the regime in which the derived asymptotic relations are valid.

References

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