超曲面叶面法揭示4D爱因斯坦引力的潜在可重归一性

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本文探讨了一种新颖的方法，即超曲面叶面法，用于重新归一化爱因斯坦广义相对论的阿奇博尔德-迪雷克（ADM）形式化表述。在Lagrangian框架下，作者通过运用四维流形上的微分同构和三维测量零的对称性，实现了对时空度量的显著去非物理化处理。这一过程引入了一个与哈密顿形式中哈密顿约束相对应的约束条件，这个约束条件的满足表明四维动力学可以有效地在三维超曲面层面上进行刻画。在传统的ADM分解中，完整 gauge 固定和几何的超曲面划分起着关键作用。这种方法暗示了4D爱因斯坦引力的理论可能具有潜在的可重归一化性，这是一个重要的发现，因为如果4D引力不可重归一化的问题源自于非物理场的存在，那么这可能揭示出一个理论基础层面的误解。研究者I.Y. Park从Hanyang University和Philander Smith College分别进行工作，他们提出这一方法适用于平坦背景下的量子化处理，如Schwarzschild解。然而，当应用到更复杂的、明确时间依赖的背景时，这种方法可能会面临一些挑战和限制。这包括可能遇到的额外复杂性，以及如何确保在实际应用中保持理论的有效性和一致性。作者强调，他们的工作提供了一个新的视角，但进一步的理论验证和实际应用可能需要深入的数学分析和物理检验。这篇文章不仅深化了我们对引力理论的理解，还提出了一个有潜力解决长期困扰理论物理学家的重归一化问题的新途径。这不仅对基本粒子物理学，也对宇宙学和引力波研究具有深远的影响，因为它可能推动我们朝着更精确、更完整地理解引力现象的道路上前进。

Eur. Phys. J. C (2015) 75 :459 Page 3 of 11 459

There exists a gauge, called the radiation gauge [29,30],

in which all of the unphysical ﬁelds are removed. The gauge-

ﬁxing procedure that we adopt keeps only the physical

degrees of freedom as in the radiation gauge, but is different

from the radiation gauge in that it leads to effective reduction

of the dynamics onto the hypersurface. The hypersurface foli-

ation approach

in this work provides a convenient arena for

examining the renormalizability after removal of the unphys-

ical degrees of freedom. Let us consider the 3 + 1 splitting

of the 4D Einstein–Hilbert action. Compared with the exist-

ing covariant approach, the ADM formulation employed in

this manuscript has two advantages: ﬁrstly, the ADM formu-

lation is effective in organizing the degrees of freedom for

easy isolation of the unphysical degrees of freedom. In other

words, the formulation readily identiﬁes the non-dynamical

ﬁelds thereby setting the stage for their removal. Secondly,

the ADM splitting brings out the utility of the measure-zero

gauge symmetry for removal of the non-dynamic ﬁelds. This

feature will play a crucial role in Sect. 2.

The rest of the paper is organized as follows. In Sect. 2,

we discuss the removal of unphysical degrees of freedom.

After starting off by recalling the gauge-ﬁxing procedure of

a YM type gauge theory, we note that it should be possible to

gauge-ﬁx the lapse function and shift vector by the measure-

zero diffeomorphisms after ﬁxing the bulk gauge symmetry

(the de Donder gauge will be adopted for the bulk ﬁxing). As

is well known (see, e.g. [35]), the choice of the lapse and shift

is arbitrary, hence gauge-ﬁxing the lapse and shift should be

a legitimate procedure in any case. What is important is that

it should be possible to gauge-ﬁx them by using the measure-

zero gauge symmetry, not the bulk ones. The result of these

gauge-ﬁxings is the projection of 4D dynamics onto the 3D

hypersurface: the 3D system has two physical degrees of free-

dom inherited from 4D. (The reduction is limited to a pure

Einstein system; a matter ﬁeld, if present, will not be reduced,

at least not in any simple way.) By invoking the logic of [5,20]

one arrives at renormalizability. This will be pointed out in

Sect. 3, in which we will also comment on precisely what

physics the reduced theory should describe. The procedure

should be viewed as a generalization of the holography idea

of [36].

A ﬂat spacetime will be considered throughout. It

should be possible to apply the procedure to a Schwarzschild

black hole background with relatively minor modiﬁcations.

However, it is not clear whether the procedure can be applied

to a more complex background such as an explicit time-

The hypersurface foliation approach combined with explicit dimen-

sional reduction has been fruitful [31–34] Unlike these works, no

explicit dimensional reduction is carried out here: the projection onto

3D is dictated by removal of unphysical degrees of freedom through the

measure-zero diffeomorphism. Thus the reduction is “spontaneous”.

’t Hooft observed the Holography in the black hole context. If what we

propose here is true, gravity theory itself has the holographic property

through a lower dimensional gravity theory.

dependent black hole background, and this is potentially a

limitation of the procedure. For the case of globally hyper-

bolic spacetimes, the reduction can easily be understood

from a different and more mathematical perspective [37].

(The condition of global hyperbolicity is not strictly required,

though.) This result is summarized in Sect. 3.2. In the conclu-

sion, we comment on several issues such as recovery of the

4D covariance in the present context or the fully 4D-covariant

formulation of the quantization. (Progress has been recently

made in [23].) In the Appendix, we illustrate the gauge-ﬁxing

dependence of a renormalization procedure by taking a sys-

tem of metric coupled with a scalar. The sole purpose of

considering this system is to demonstrate the dependence:

we consider only the pure gravity system in the main body.

2 Removal of unphysical degrees of freedom

It will be useful for what follows to recall the quantization

procedure in Maxwell’s theory. The vector ﬁeld has four com-

ponents to start with but only two of them are physical degrees

of freedom. The system has gauge symmetry; it reduces the

number of degrees of freedom to three. The time compo-

nent is non-dynamical, leading to the further reduction of the

number of physical ﬁelds to two. Let us consider temporal

gauge to be speciﬁc. It turns out that temporal gauge does not

entirely use up the gauge freedom but leaves gauge symme-

try associated with the hypersurface of a ﬁxed time [38]. For

the perspective of our gravity analysis, what is important is

that the non-dynamical time component can be gauge-ﬁxed

without using the full bulk symmetry but instead by using

measure-zero lower dimensional gauge symmetry. Below we

will show that there is an analogous procedure in general rel-

ativity.

As in the Maxwell case, it is the close conspiracy between

non-dynamism and gauge symmetry that brings complete

removal of the unphysical degrees of freedom. In due course

of the analysis below, the lapse function n will get ﬁxed to

n = 1. (This choice is suitable because we are consider-

ing expansion of the theory around a ﬂat spacetime to be

speciﬁc.) This introduces a constraint that corresponds to

Hamiltonian constraint of the Hamiltonian formulation. As

we will see, the constraint can be solved, and its solution

implies, among other things, the effective projection of the

4D dynamics onto the 3D hypersurface.

The gauge-ﬁxings of the measure-zero symmetries will

be carried out following the spirit of section 15.4 of [39]

in which the ﬁrst class constraint was eliminated by ﬁxing

the corresponding symmetry. In essence, what we do here is

ﬁx the gauge and explicitly solve the resulting constraints.

Therefore the procedure does not introduce any ghosts at the

bulk level. (In [39], quantization in the axial gauge was ana-

lyzed, and it was shown that the axial gauge quantization does

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