Electric Dipole Paradox: Question, Answer, and Interpretation Frank Wilczek January 16, 2014 Abstract Non-vanishing electric dipole moments for the electron, neutron, or other entities are classic signals of P and T violation. Electric dipole moments of water molecules are, on the other hand, a relatively mundane working concept in chemistry. The distinction is instructive, and says something profound about the meaning of elementary particles. It illustrates the importance of energy gaps, in allowing the Quantum Censor to do its work. Measurements sensitive to possible electric dipole moments for electrons, muons, neutrons, and various atoms and molecules are among the most sensitive null experiments in physics. They put powerful limits upon the size of those moments, constraining them to be many orders of magnitude below the dimensional analysis limits one derives by multiplying the characteristic overall sizes of the objects times their charge (or, in the case of neutral objects, the charges of their constituent electrons, nuclei, or quarks). For example, recent experiments constrain the electric dipole moment of the electron, defined via the interaction energy by E = d e E (1) d e 10.5 10 28 e cm. (2) This is much smaller than simple dimensional analysis, whether based (as it should be) on the electron s quantum radius, or Compton wavelength r e Compton h m e c 3.8 10 11 cm. (3) 1
or on its classical radius r e classical e2 mc 2 2.8 10 13 cm. (4) which is smaller by a factor of the fine-structure constant α. Fundamental principles of quantum mechanics and relativity do not preclude the existence of such moments, and in fact they are calculated to arise (with tiny magnitudes) in the Standard Model. Many hypothetical extensions of the Standard Model predict electric dipole moments that might be observed in achievable experiments, and that circumstance defines an important, active area of current research. The extraordinary suppression of the moments, compared to dimensional expectations, is explained by the approximate invariance of low-energy physics under spatial P and, especially, time reversal T symmetry. On the other hand there is a simple, well-known picture which suggests that water molecules, for example, should have an electric dipole moment, because their charge distribution is far from uniform. And in chemistry texts one finds tables of measured molecular electric dipole moments, that compare well with calculated models of charge distribution. What s the difference? After all, when viewed from afar the water molecule is just another particle. And how can we be sure, a priori, that the dipole-less particles don t have water molecule-like internal structure? (In fact the neutron does have plenty of poorly understood internal structure.) We d better be able to explain the difference precisely, if we re to reconcile the fantastic accuracy of the null experiments with the relatively mundane, seemingly straightforward non-null experiments! In the first part of this short note I ll answer that question directly. In the second part I ll put the answer into a larger context, using it to exemplify the profound concept of Quantum Censorship. Electric Dipoles, Angular Momentum, and Symmetry The interaction energy of a particle with a constant electric field E can depend upon its spin S according to the Hamiltonian H = d E (5) which can be considered the definition of the dipole operator d. We want to evaluate this for our particle of interest. Now the state of a free particle is completely specified by the orientation of its spin degree of freedom. (We ll return to that key point below.) We can choose a basis labelled by angular 2
momentum along the ẑ axis, as usual, for our particle of total spin s. Then our task, in evaluating the Hamiltonian expectation value of the Hamiltonian (5), is to evaluate the matrix elements s, m d s, m d m m (6) This quantity must behave as a vector under proper rotations, in order for the Hamiltonian in Eqn. (5) to be rotationally invariant. According to the rules for combining angular momenta, there is exactly one way, up to an overall proportionality constant, to construct a quantity of the type ( V ) m m that transforms as a vector when rotations act on the spin indices m, m (Wigner-Eckart theorem) 1. On the other hand, there is a quantity of this kind that is intrinsic to the particle, namely the expectation value of the angular momentum operator, i.e. J m m s, m J s, m (7) Thus rotational symmetry, expressed mathematically through the Wigner- Eckart theorem, tells us that s, m d s, m = γ s, m J s, m (8) where γ is a scalar constant. Now J behaves as an improper vector (also called pseudovector or axial vector) under spatial inversion P, and also reverses sign under time reversal T. One the other hand E is a proper vector, and does not change under T. Thus an interaction of the kind we ve looking for s, m H s, m = γ E s, m J s, m (9) changes sign under P and T, unless γ = 0. This reasoning takes us where we wanted to go, for electrons and neutrons, but it seems to prove too much, because we need to allow for the non-zero moment of water molecules. What s missing? The point is that the Hamiltonian in Eqn. (5) is an operator, and in evaluating its expectation value we must respect that fact. In the preceding argument, we assumed that we could focus on the degenerate ground states of our particle, and ignore any other states. We can test the consistency of that assumption, by treating the effect of other states in perturbation theory. We can expect that if the perturbation is small enough, and the energy 1 Trivial exception: for s = 0, there is no way. 3
gap to higher states large enough, that effect will be small. In principle, for small enough E, any finite energy gap will be sufficient, but if the energy gap is small, we can get a different result. When the perturbing matrix elements n, s n, m d E 0, s 0, m (10) connecting the ground state (labelled 0) to higher-energy states (labelled n) with are comparable to or greater than the diagonal, intrinsic splittings E n E 0 then we must abandon ordinary perturbation theory and diagonalize the full problem, at least for those states (degenerate perturbation theory). The full notation, when we spell all this out concretely, exhibits distracting complexity, so for present purposes I will retreat to a simpler model problem, that exhibits the essence of the matter. Consider two states with an intrinsic splitting S and an off-diagonal, perturbing interaction, leading to the matrix Hamiltonian ( ) S (11) 0 with the eigenvalues E ± = S 2 ± ( S 2 )2 + 2 (12) For small, the lower eigenvalue varies quadratically in, as while for large E E ± In this set-up, we are modeling S 2 2S (13) S ± (14) E d E J (15) For very small electric fields, we the energy varies quadratically in the field, so we have no true (linear) electric dipole moment, but at larger fields eventually we find linear dependence, so the spectrum mimics what we d get for a true moment. Note, however, that the result depends on the absolute value of the perturbation, so it is perfectly consistent with P and T symmetry! Now in the application to water molecules, there will be rotational levels, formed by taking different superpositions of states with the molecule oriented 4
in different directions, to form angular momentum eigenstates. But the splitting among those levels, however, is tiny. So although strictly speaking the electric dipole moment in the ground state, with zero angular momentum, vanishes, linear behavior of the energy in E takes over at quite small values of the electric field. It is the coefficient governing that proportion which experimentalists ordinarily report as the electric dipole moment. Quantum Censorship A semi-classical model of the water molecule would have it as an isosceles triangle, with the oxygen atom at the vertex of a 104.45 angle. Negative charge accumulates at the oxygen, leaving positive charge toward the hydrogens, generating an electric dipole moment. Now when we quantize this semi-classical object, to form states of definite angular momentum, we superpose states corresponding to rotating that semiclassical framework into all possible orientations, with appropriate weights. To get spin 0 we superpose them all with equal weight, for higher spins we weight with appropriate spherical harmonics. The different rotational states will be split, and form a non-trivial spectrum. This is the quantum-mechanical space that replaces the classical state space of degenerate states, with the framework in all possible orientations. If we probe the spin 0 ground state gently, using perturbations whose magnitude is small compared to the energy gap, then the molecule remains in a unique state, betraying no direct hint of the original semi-classical degeneracy. Or, to put it more vividly, its internal structure has been censored. The same sort of reasoning applies to neutrons. We know that they have complex internal structure, when described in terms of the microscopic degrees of freedom (quarks and gluons) in quantum chromodynamics (QCD). Yet for many purposes, including the measurement of potential electric dipole moments and conventional nuclear physics, we can treat them as elementary particles. The point is that in this case the energy gap is quite large by the standards of probing fields, or even nuclear binding energies it is a typical QCD scale, of order 100 million electron volts. Similar, if electrons have internal structure for example, if they are secretly strings the energy gap must be enormous, on empirical grounds, and it will require extremely violent perturbations to reveal that internal structure. Quantum censorship is a general mechanism whereby complexity is hidden and discreteness emerges from continuity. Many features of the physical 5
world can only be understood with its help. It is a profound, yet underappreciated, qualitative implication of quantum theory. 6