A problem descending from H. Yukawa's 1948 letter 'Possible Types of Nonlocalizable Fields' (Prog. Theor. Phys. 3, 452). Yukawa proposed to replace point fields by 'nonlocalizable' fields -- operator functions of the noncommuting pair (x,p)(x, p), equivalently bi-local fields U(X,r)U(X, r) satisfying the reciprocal commutation relations [x[x,U]]=−λ2U[x[x,U]] = -\lambda^2 U and [p[p,U]]=−m2c2U[p[p,U]] = -m^2 c^2 U, so that a fundamental length (particle radius) enters as the quantity reciprocal to the mass. Of the three items the letter left open, two were resolved by Yukawa himself in 1949-1950: the massive case (in four dimensions, following a remark of Pais, superseding the five-dimensional route) and the spin-1/2 case (non-local scalar, vector and spinor free fields of arbitrary mass and radius). What survives is the third item and the heart of the program: a consistent theory of two or more mutually interacting nonlocalizable fields. Yukawa's 1950 S-matrix attempt was unitary and Lorentz invariant but provably could not satisfy convergence (finiteness of amplitudes) and correspondence (the λ→0\lambda \to 0 local-field limit) simultaneously; Yennie's analysis concurred, and the modern continuation (bi-local and noncommutative-spacetime realizations) still reports the same causality/unitarity obstruction. The problem -- construct such an interacting theory, or establish a no-go obstruction within the framework -- remains open.

Background

Local quantum field theory assumes that fields are point functions of spacetime, ϕ(x)\phi(x). In the 1930s-40s many physicists attributed the ultraviolet divergences of quantum electrodynamics and meson theory to this assumption: if elementary particles have finite spatial extent, the singular short-distance behavior of point fields should be cut off by a fundamental (universal) length λ\lambda. The difficulty is to introduce λ\lambda without picking a preferred inertial frame: a naive smearing of a field over a spatial region breaks Lorentz invariance. One proposed answer was Born's reciprocity principle, which treats position and momentum symmetrically, taking the field to live on the combined position-momentum space rather than on spacetime alone.

Following this line, Yukawa developed in 1947-48 the concept of a nonlocalizable field: the field operator UU is not a function of the coordinates alone but an operator function of the noncommuting pair xμx_\mu (space-time coordinate operators) and pνp_\nu (space-time displacement operators) with [xμ,pν]=iℏδμν[x^\mu, p_\nu] = i\hbar\delta^\mu{}_{\nu}. In a representation in which the xμx_\mu are diagonal, UU becomes a matrix whose elements can be written as a bi-local function U(Xμ,rμ)U(X_\mu, r_\mu) of the center Xμ=(xμ′+xμ′′)/2X_\mu = (x'_\mu + x''_\mu)/2 and the relative coordinate rμ=xμ′−xμ′′r_\mu = x'_\mu - x''_\mu -- the field-theoretic image of an extended particle. Reciprocity is imposed through paired commutation relations, [xμ[xμ,U]]=−λ2U[x_\mu[x^\mu, U]] = -\lambda^2 U and [pμ[pμ,U]]=−m2c2U[p_\mu[p^\mu, U]] = -m^2 c^2 U, so that the radius λ\lambda enters as the quantity reciprocal to the mass mm. In the 1948 letter, Yukawa exhibited two types of (scalar, neutral, zero-mass) nonlocalizable fields: one built from the real canonical combinations ημ=xμ/λ+λpμ/ℏ\eta_\mu = x_\mu/\lambda + \lambda p_\mu/\hbar, expanded in plane waves exp⁡(ikμημ)\exp(ik_\mu \eta^\mu) over null vectors kμkμ=0k_\mu k^\mu = 0, and one built from oscillator-type complex combinations ξμ=(xμ/λ−iλpμ/ℏ)/2\xi_\mu = (x_\mu/\lambda - i\lambda p_\mu/\hbar)/\sqrt{2}. He noted that the massive case had been indicated to require a five-dimensional space (per the preceding letter), and left the spinor field and the coexistence of two or more interacting nonlocalizable fields for later investigation.

The follow-up primary literature settles the kinematic part. The massive case was reformulated in four dimensions after a remark of Pais, and Yukawa's 1949-50 papers constructed the full free-field theory: non-local scalar, vector, and spinor fields, each describing an assembly of particles with a definite mass mm, a definite radius λ\lambda, and spin 00, 11, or 1/21/2, classified by four invariants (mass, radius, magnitude of internal angular momentum, and a fourth constant without direct physical meaning). Two structural facts about the free theory matter for what follows. First, Fierz showed that the free non-local fields are mathematically equivalent to (reducible) assemblies of local fields of arbitrary spin, with the relative-coordinate space interpretable as spin space -- so the entire physical content of the framework must reside in the interaction. Second, because a Lorentz-covariant extended object necessarily extends in timelike as well as spacelike directions, an initial-value (Schrodinger-type) formulation is in general unavailable, which pushes the interaction problem toward integral formulations such as Heisenberg's S-matrix scheme (the scattering operator relating the remote past to the remote future, used to characterize interactions without a Hamiltonian evolution).

In Part II of the 1950 series Yukawa attempted exactly this: an interaction defined through an invariant, unitary S-matrix built from integrals of products of non-local field operators. The attempt satisfied Lorentz invariance and unitarity but failed on the remaining two requirements, and this is where the program stopped. The question has since been revisited in the bi-local re-interpretation of Yukawa's fields as relativistic two-body bound-state models, and in a modern realization of Yukawa-type nonlocal fields out of noncommutative spacetime (coordinates obeying a Lie algebra, e.g. [x^0,x^i]=iκ−1x^i[\hat{x}^0, \hat{x}^i] = i\kappa^{-1}\hat{x}^i), where interacting nonlocal field equations of sinh⁡((κμ)−1∂2)\sinh((\kappa\mu)^{-1}\partial^2) type yield finite one-loop diagrams but contain complex poles that may spoil unitarity, and the timelike extension of the fields conflicts with causality. Adjacent modern programs (for example ghost-free infinite-derivative field theories, whose kinetic operators are entire functions of the wave operator (the d'Alembertian), so that a fundamental length enters through infinitely many derivatives rather than through new particle-like poles, and noncommutative field theory) actively study the same trade-off cluster -- nonlocality versus causality, unitarity and ultraviolet behavior -- but with different nonlocality structures, and none constructs an interacting theory of reciprocity-type nonlocalizable fields.

Problem Statement

As it survives from Yukawa's 1948 letter, the problem is the interaction half of the nonlocalizable-field program. Construct a consistent quantum theory of two or more mutually interacting nonlocalizable fields -- fields UU that are operator functions of the noncommuting pair (xμ,pν)(x_\mu, p_\nu), equivalently bi-local functions U(X,r)U(X, r), satisfying the reciprocal commutation relations

[xμ[xμ,U]]=−λ2U,[pμ[pμ,U]]=−m2c2U[x_\mu[x^\mu, U]] = -\lambda^2 U, \qquad [p_\mu[p^\mu, U]] = -m^2 c^2 U

with fundamental length λ\lambda (particle radius) reciprocal to the mass mm, for arbitrary masses and spins (free scalar, vector and spinor fields of this type being available since 1950). The interaction may be introduced by any means internal to the framework: a Lagrangian operator for the coupled system, an S-matrix definition, or a modification of the reciprocal commutation constraints themselves. 'Consistent' is to be understood in Yukawa's own 1950 formulation of the requirements: the theory must simultaneously satisfy (i) Lorentz invariance; (ii) unitarity; (iii) convergence -- finiteness of the amplitudes, which is the purpose of introducing the fundamental length; and (iv) correspondence -- reduction to the results of the usual local field theory in the limit λ→0\lambda \to 0. Yukawa's 1950 S-matrix attempt satisfied (i) and (ii) but demonstrated that (iii) and (iv) cannot be fulfilled simultaneously within that S-matrix construction. The question is whether any construction within this framework can satisfy all four requirements simultaneously -- including the subsidiary question of whether and how the interaction must modify the reciprocal commutation constraints -- or whether a no-go obstruction exists.

References

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