II-2. Nuclear Physics (Theory)

RIKEN Accel. Prog. Rep. 54 (2021)

RIKEN Accel. Prog. Rep. 54 (2021)

Non-relativistic expansion of Dirac equation by the reconstituted
Foldy-Wouthuysen transformation†
Y. X. Guo∗1,∗2 and H. Z. Liang∗2,∗1
In the past few decades, the density functional theory(DFT) has been successfully applied in both the nonrelativistic and relativistic frameworks to describe theground-state and excited-state properties of thousandsof nuclei in a microscopic and self-consistent manner.
However, the connection between these two frameworks
remains unclear. The non-relativistic expansion of theDirac equation is considered to be a potential bridgeconnecting these two frameworks.1,2)
Recently, the reconstituted similarity renormalization group (SRG) method was proposed.3) Compared toconventional methods, the reconstituted SRG method
not only showed a much faster convergence, but also
yielded single-particle densities ρv (r) = ψ † (r)ψ(r) thatare closer to the exact values. Moreover, for the singleparticle scalar densities ρs (r) = ψ † (r)γ0 ψ(r), the γ0matrix should be transformed in the same manner as
the Dirac Hamiltonian. However, the original Dirac
Hamiltonian is operated with inﬁnite steps of unitarytransformations in both the conventional and reconstituted SRG methods. Therefore, the calculations of
single-particle scalar densities are not trivial in SRG
methods.
Meanwhile, with the consideration of the strong
scalar potential, we further applied another well-known
technique, the Foldy-Wouthuysen (FW) transformation, to the general cases in the covariant DFT andperformed the corresponding expansion up to the 1/M 4order.4) With the present FW transformation, all theunitary transformations involved hold explicit forms.
On this basis and with inspiration from the reconstituted SRG method, we developed the reconstituted FWtransformation. By replacing the bare mass M with theDirac mass M ∗ and deﬁning the corresponding new operators, the contributions related to the higher powersof the quotient of the scalar potential and the bare mass
are absorbed into the lower orders.
As a step forward, the so-called picture-change error,5,6) i.e., the diﬀerence between the densities calculated in the Schrödinger picture and those obtained in
the Dirac picture, has also been investigated throughcalculations starting from basic ﬁeld operators. Afterconsidering such picture-change errors, i.e., the relativistic corrections, both the single-particle vector andscalar densities become much closer to the exact results.
In addition, the diﬀerence between ρv and ρs originatesfrom the 1/M ∗ 2 order, as does the relativistic correc†∗1∗20.202d5/2

Exact
Without corr.
21/M* corr.
31/M* corr.

-3-3

ρv - ρs (10 fm )

0.15

0.10

0.05

0.00
246810r (fm)Fig. 1. Diﬀerence between the single-particle density andthe single-particle scalar density for the neutron 2d5/2state of nucleus 208 Pb.

tion ∆ρ. Therefore, the relativistic corrections also playa crucial role in the diﬀerence between the vector andscalar densities.
By taking the 2d5/2 state of 208 Pb as an example, thediﬀerence between the single-particle density and thesingle-particle scalar density is shown with the blacksolid line in Fig. 1. The result without any relativistic correction is shown with the olive dashed line, while
the results calculated with the consideration of relativistic corrections up to the 1/M ∗ 2 and 1/M ∗ 3 orders areshown with the blue dotted and red dash-dotted lines,
respectively. It can be seen from Fig. 1 that the diﬀerences between the results without the relativistic corrections and the exact values are systematic. There are
diﬀerences in the positions of peaks and nodes as wellas the amplitudes. In contrast, the results with the relativistic corrections remarkably reproduce the behaviorof the exact values, particularly in terms of the positionsof peaks and nodes.
Based on the above discussions, the reconstituted
FW transformation is a promising method to connect
the relativistic and non-relativistic DFTs for futurestudies.
References

1) P. G. Reinhard, Rep. Prog. Phys. 52, 439 (1989).
2) M. Bender, P. H. Heenen, P. G. Reinhard, Rev. Mod.
Phys. 75, 121 (2003).
3) Y. X. Guo, H. Z. Liang, Phys. Rev. C 99, 054324 (2019).
4) Y. X. Guo, H. Z. Liang, Chin. Phys. C 43, 114105 (2019).
5) M. Douglas, N. M. Kroll, Ann. Phys. (N. Y.) 82, 89(1974).
6) B. A. Hess, Phys. Rev. A 33, 3742 (1986).

Condensed from the article in Phys. Rev. C 101, 024304(2020)
Department of Physics, The University of Tokyo
RIKEN Nishina Center- 50 -1