构建dS/CFT对偶下的高自旋重力线性化量子化

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本文探讨了"De Sitter因果斑中线性化的高自旋重力的全息量化"这一主题，聚焦于最小A型高旋转引力与自由Sp(2N)矢量模型之间的dS / CFT对偶。在这个研究中，作者将整个时空视为"椭圆"的de Sitter空间dS4 / Z2，即在其中两个对映点被识别。他们应用了一种创新技术，源于arXiv:1509.05890论文，该技术能够从欧几里得CFT的边界2点函数中直接提取出dS4 / Z2可观测区域中线性化体理论的量子力学交换子（或泊松括号），从而构建出线性化体理论的洛伦兹变换器。作者使用高自旋协变语言处理整个高自旋多重峰，这是为了为未来包含更复杂的 Bulk (体) 交互提供一个有前景的框架。这种方法不仅有助于深化我们对dS / CFT对偶的理解，而且构建哈密顿结构的过程对于理解高自旋理论本身具有重要意义。然而，这样做会牺牲部分对称性，从完整的de Sitter群（包括其高旋转扩展）降低到可观测区域的对称性组。尽管边界场论在本文中起关键作用，但这些结果同样适用于边界粒子力学的表述。体场通过二次量子化的方式从这种边界机制中产生，这是一种非传统的视角，有助于进一步探索高自旋理论的量子性质。值得注意的是，这项工作发表于《Journal of High Energy Physics》(JHEP) 2018年11期，是开放获取的，这意味着研究成果可供全球科研人员无限制访问。接收日期为2018年9月25日，接受日期为11月1日，最终出版日期为11月6日。总体而言，这篇文章在高自旋重力的理论发展和dS / CFT对偶的深入理解上做出了重要的贡献，对于理论物理学家来说，它提供了一个独特的工具来处理复杂且精细的量子效应。

JHEP11(2018)033

mechanics. Using the symplectic form (1.5), we will obtain explicit expressions f or the bulk

commutators:



hu|

F |¯ui,

hv|

F |¯vi



4πiN

∞

−∞

dα

α−1

α+1



v−

√

αu



+δ



√

αu



+δ



v−

√

α ¯u



+δ



√

α ¯u



(1.7)

as well as for the Hamiltonian:

H =

4π



i¯u

∞

−∞

|α|dα

(α + 1)

√

α u



√

α i¯u

. (1.8)

One can also work in a bulk frequency basis, arrange the bulk ﬁel ds i nto creation and

annihilation operators, and rewr i t e the Hamiltonian (

1.8) in normal or der i n g. This will

be accomplished in section

4.5.2. The HS symmetry of our construction is analyze d in

section

4.6. Secti on 5 is devoted to discussion and outlook.

2 The Wigner-Weyl transform and doubled phase space

The Wigner-We y l transform [

25] between functions on phase space and operators on Hilbert

space plays a crucial role in this paper. I n this section, we review three useful deﬁnitions

of this transform. Our focus is on quantum systems that can be obtained by quantizing

a linear phase space. One such system is linearized HS gravity in the 4d bulk. Another

example, as we’ll see in se ct i on 4.3, is a free massless particle on the 3d boundary.

2.1 Symmetric ordering and the Moyal star product

Consider a linear phase space, parameterized by 2N-dimensional vectors ξ

, where N is

the (possibly inﬁnite) number of degrees of freedom. The phase space is equipped with a

symplectic form:

Ω =

Ω

dξ

, (2.1)

with inverse (Ω

−1

)

. The linear canonical transformations that preserve this form make

up the symplect i c group Sp(2N). Th e Heisenberg commutators are given by:

[

] = i(Ω

−1

)

. (2.2)

Any operator on the system’s Hi l bert space can be construc te d as a sum of products of

the fu ndame ntal operators

. Given some ordering convention for such products, we can

treat their coeﬃcients as the Taylor expansion of a f unc ti on on the classical phase space.

Thus, any orderi ng convention yields a 1-to-1 map between operators

Ψ on the Hilbert

space and functions Ψ(ξ

). The only ordering convention that prese r ves the full Sp(2N)

group is Weyl ordering, in which the classi c al product ξ

. . . ξ

is mapped to the fully

symmetrized operator product:

. . . ξ

←→

. . .

)

. (2.3)

– 7 –

JHEP11(2018)033

To see that (2.3) is indeed uniq ue, note that the l.h.s. is completely symmetric in its in-

dices, whi l e any Sp(2N)-symmetric corrections to the r.h.s. would have to be constructed

using the antisymmetric Ω

. Operator orderings other than (

2.3), such as normal order-

ing, always involve some Sp(2N)-breaking extra st r uc tur e on the phase space, such as a

decomposition into positive and negative frequencies. The map (

2.3) be tween operators

and phase space functions Ψ(ξ

) is our ﬁrst deﬁnition of the Wigner-Weyl transform. The

transform deﬁnes a non-commutative pro duc t on phase spac e functi ons , which reproduces

the product of the corres ponding oper at or s. This is the Moyal star product ∗, deﬁned by

the following equivalent formulas:

∗ ξ

= ξ

(Ω

−1

)

;

Φ(ξ) ∗ Ψ(ξ) = Φ(ξ) exp

(Ω

−1

)

←−−

∂

∂ξ

−−→

∂

∂ξ

Ψ(ξ)

= 2

′

′′

Φ(ξ + ξ

′

) Ψ(ξ + ξ

′′

) e

−2iΩ

′I

′′J

(2.4)

In the last formula, we use the phase space measure derived from Ω

ξ = ±

Ω

. . . Ω

2N −1

dξ

. . . dξ

2N −1

dξ

N!(2π)

, (2.5)

where we inserted a slightly nonstandar d 1/(2π)

factor, eﬀectively swi tching from units

of ~ to uni t s of h. This will prevent any 2π factors in our Fourier and Gaussian integrals.

In particular, wi t h this deﬁnition, we have:

′

iΩ

′I

≡ δ(ξ) ;

ξ δ(ξ) = 1 . (2.6)

Note also that we left the sign in (

2.5) undetermined. Normally, this sign is arbitrary: one

always has the compl em entary freedom to choose the order of integration l i m i ts , so that e.g.

the integral of 1 comes out positive. However, the freedom to choose the sign of d

ξ will

be important for us in section

4, where we’ll consider twistors as a complex phase spac e.

Finally, Hermitian conjugation of an operator

Ψ →

†

is re pr es ented by ordinary

complex conjugation of the phase space function Ψ(ξ) →

Ψ(ξ). Operator products are

reversed by Hermitian conjugation, and the star product (

2.4) transforms accordingly as

Φ(ξ) ∗ Ψ(ξ) →

Ψ(ξ) ∗

Φ(ξ).

2.2 Doubled phase space and the Hilbert spac e of operators

For our second deﬁnition of the Wigner-Wey l transform, we conside r a doubling of the

phase space ξ

into a 4N-dimensional space with coordinates (ξ

, ξ

−

), equipped with the

symplectic form:

Ω

double

Ω



dξ

− dξ

−

dξ

−



. (2.7)

– 8 –

JHEP11(2018)033

In other words, under Ω

double

, the ξ

form an isomorphic copy of the original phase space,

while ξ

−

form a time-reversed c opy. We will later see this picture realized geometrically in

de Sitter space.

At the quantum level, the doubled phase space (

2.7) corresponds to a doubled Hilbert

space of the form H ⊗ H

∗

. The anti-isomorphism (ξ

, 0) ↔ (0, ξ

−

) between the “upright”

and time-reversed copies becomes the complex conjugation |ψi ↔ hψ| between bras and

kets. To see why this map should be anti-linear, note that ﬂipping Ω

→ −Ω

in t he

commutator (

2.2) can be canceled by ﬂipping i → −i. Thus, we are led to two equivalent

interpretations of the space H ⊗ H

∗

. On one hand, it is the Hilbert space of states ||Ψii

of the doubled system (

2.7). On the other hand, it is of course the space of ope r ator s

on the original Hilbert space H. The inner product on the doubled Hilbert space can be

expressed via the operator algebra on H as:

hhΨ||Φii = tr(Ψ

†

Φ) . (2.8)

Explicitly, one often likes to deﬁne states as wavefunctions on some “conﬁguration

space”, i.e. on a polarization of phase space. In the original phase space ξ

, we do not

have a preferred choice of polariz ati on. However, in the doubled phase space (2.7), we do!

Indeed, let us deﬁne the even and odd combinations:

even

+ ξ

−

; ξ

odd

− ξ

−

. (2.9)

The doubled sympl e ct i c form (2.7) now reads:

Ω

double

= 2Ω

dξ

even

dξ

odd

. (2.10)

Thus, either ξ

even

or ξ

odd

can serve as a preferred polarization of the doubled phase space,

with the other set then serving as the canonically conjugat e “momenta”.

One can now express states ||Ψii on the doubled Hil bert space H⊗H

∗

as wavefunctions

in either the ξ

even

basis or the ξ

odd

basis:

Ψ(ξ

) = hhξ

even

= ξ

||Ψii; (2.11)

Ψ(ξ

) = hhξ

odd

= ξ

||Ψii. (2.12)

Our notat i on here emphasizes that, si nc e all the copies of our original phase space are

isomorphic, the same coordinates ξ

can denote a point either in the ξ

even

subspace, or

in the ξ

odd

subspace. The se two options can be visual i ze d by putting t he classical history

described by ξ

in the “upright” phase space copy ξ

, and then putting eithe r +ξ

or −ξ

in the “time-reversed” copy ξ

−

Combining th e conﬁguration basis (

2.11) for states in H ⊗ H

∗

with the interpretation

of these states as operat or s on H, we arrive at a map between phase space functions Ψ(ξ

)

and operators

Ψ:

Ψ(ξ

) ←→ ||Ψii ←→

Ψ . (2.13)

As we’ll see in the next subsec ti on, this map is once again the Wigner-Weyl transform

deﬁned by (

2.3). We will return to the role of

Ψ(ξ

) in section 2.4.

– 9 –

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