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首页二维共形场论中扰动的路径积分复杂度研究
"扰动CFT的路径积分复杂度" 这篇学术文章主要探讨的是二维共形场理论(CFT)在受到相关算子扰动时的路径积分优化方法。路径积分是量子场论中的一种重要计算工具,它通过整合所有可能的量子态来描述系统的演化。在本文中,作者们提出了一个针对这类扰动CFT的优化机制,其目标是通过最小化依赖于度量和相关耦合的路径积分复杂度函数来寻找最优路径。 作者们首先在自由场理论的框架内进行了计算,这是一个相对简单的理论模型,可以用来理解更复杂的相互作用场论。他们利用这些计算结果以及场论中的Renormalization Group (RG) 流理论作为证据,说明优化机制的工作原理。RG流是一种描述物理系统在不同尺度下行为变化的理论,它可以用来理解和预测在不同的能量尺度上,理论的性质如何变化。 在具体操作中,作者们对最优度量进行了扰动计算,发现这个最优度量与AdS/CFT对应关系中的双曲度量在标量场扰动下的时间片相一致。AdS/CFT对应是弦理论和量子场论之间的一个强大连接,其中Anti-de Sitter(AdS)空间的几何特性与共形场论的复杂性有直接关联。 最后,作者们估算了相关扰动对复杂性的贡献。复杂性在量子信息科学和黑洞物理学中是一个重要的概念,它衡量了执行特定量子计算的难度。在CFT的背景下,这涉及到对系统状态的复杂编码和处理。 该研究深化了我们对扰动CFT中路径积分复杂性的理解,特别是在考虑度量和耦合变化的影响时。这不仅有助于理解量子场论的基本性质,也为探索AdS/CFT对应关系提供了新的视角,进一步推动了量子引力和高能物理的研究。此外,这项工作的结果也可能对量子信息处理和计算复杂性的理论研究产生影响。
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JHEP07(2018)086
them as g(x, z) and λ(x, z) and impose the boundary conditions
g(x, ) = g
0
, λ(x, ) = λ
0
. (2.3)
At z = the metric remains same as the original metric and thus the fields are not
rescaled. So we need to impose this boundary condition which is same as what is used for
the original wavefunctional, so that after the rescaling of the metric (and the fields) the
new wavefunctional will be same as the original one.
In general, the wave functional obtained from the Euclidean path-integral on this
curved space with the position dependent coupling constants
Ψ
g(x,z),λ(x,z)
[ϕ(x)] =
Z
Y
x
Y
<z<∞
[Dϕ(x, z)] e
−S
g(x,z),λ(x,z)
[ϕ]
Y
x
δ(ϕ(x, ) − ϕ(x)), (2.4)
differs from the original one Ψ
g
0
,λ
0
[ϕ(x)] (2.1) in a nontrivial fashion. However, if we
fine tune g(x, z) and λ(x, z), then we can find non-trivial functions g(x, z) and λ(x, z)
which, in the path integral approach, give rise to wave functionals proportional to the one
computed with their boundary values Ψ
g(x,z),λ(x,z)
[ϕ(x)] ∝ Ψ
g
0
,λ
0
[ϕ(x)]. This means that
they describe the same quantum state. More precisely, for these functions we can write
Ψ
g(x,z),λ(x,z)
[ϕ(x)] = e
N[g,λ]−N[g
0
,λ
0
]
· Ψ
g
0
,λ
0
[ϕ(x)]. (2.5)
The optimization procedure can be completed by minimizing the normalization factor, or
equivalently minimizing the functional N[g, λ] with respect to g(x, z) and λ(x, z). The
position dependent metric and coupling constants which minimize N [g, λ] are written as
g
min
and λ
min
and later we will suggest that they correspond to the metric and bulk fields on
a time slice of AdS for holographic CFTs. In addition, we define the quantity N [g
min
, λ
min
]
as the path-integral complexity (denoted by C[λ
0
]) for the vacuum state in the QFT, given
by the wave functional Ψ
g
0
,λ
0
[ϕ(x)]:
C[λ
0
] ≡ Min
g(x,z),λ(x,z)
N[g(x, z), λ(x, z)]. (2.6)
It is also straightforward to extend the path-integral optimization to general excited states.
This is because once we have a path-integral description of a quantum state, which we want
to consider (e.g. inserting local operators in the middle of the path-integral) is given, in
principle, the optimization by locally deforming the metric and coupling constants can
be performed.
2.2 Interpretation
Before we go on to explicit examples, let us explain an intuitive idea behind our prescription.
First, consider a numerical computation of path-integral to calculate Ψ
g
0
,λ
0
[ϕ(x)] (2.1). We
normally fine-grain both z and x coordinate for the metric (2.2) such that the size of each
cell is given by ∆z = ∆x = . What we have in our mind here is that, we perform the
discretization of the path-integral in a way, such that each unit area square corresponds
to a tensor T
a
1
,a
2
,...,a
n
. For example, in figure 1 we can interpret each square cell as a
– 4 –
JHEP07(2018)086
tensor with 4 indices (n = 4). When two cells are attached along an edge, we contract the
corresponding indices. The path-integral is then approximated by all of such contraction of
all these tensors.
1
However, if we think about the path-integral in an early time τ → −∞,
we do not need such fine grained information of the wave functional at that time as we
path-integrate for a long time afterwards, as explained in [10, 11, 23]. This means that we
can coarse-grain the cells in the past. In order to reproduce the correct wave functional
Ψ
g
0
,λ
0
[ϕ(x)], we need to reduce the amount of coarse-graining as the time evolves, ending
up with the fined grained lattice at z = . In terms of the tensors, we have to recombine
the initial tensors and replace them by fewer numbers effective tensors. The righthand side
of the figure 1 shows this coarse grained geometry where again each of cells is interpreted
as a tensor with 5 indices (n = 5). Again contractions of all indices of these tensors
give the discretized path-integral. This coarse-graining procedure can be systematically
described by locally changing the metric and coupling constants as g(x, z) and λ(x, z) with
the boundary condition (2.3).
Next, we want to make the numerical computation of the path-integral the most effi-
cient; the process which we call the optimization of the path-integral. Notice that here we
do not want to change the dependence of the final wave functional on the field configuration
after the path-integration, which gives the constraint (2.5). We minimize the amount of
algebraic computations in a lattice regularization to obtain the correct ground state wave
functional. We argue this can be performed by minimizing the overall normalization of
wave functional, given by e
N[g,λ]
in (2.5). This is because N [g, λ] is an obvious measure
which estimates the number of path-integral operations to obtain a given quantum state.
This argument was also justified in [24] from the viewpoint of complexity of MERA tensor
networks [13, 14], which describe ground states of CFTs in two dimensions.
3 Path-integral optimization of 2D CFTs and relevant peturbations
A class of QFTs where the optimization procedure is tractable is given by two dimensional
conformal field theories [10]. Here first we would like to briefly review an explicit optimiza-
tion procedure for two dimensional CFTs, focusing on the vacuum state. Refer to [23] for
more detailed computations as well as generalization to excited states. In addition, we will
present an argument which provides an extra support of our procedure. Next we turn to
the main aim of this work i.e. the path-integral optimizations of QFTs defined by relevant
perturbations of two dimensional CFTs.
3.1 Path-integral optimization of 2D CFTs
For conformal field theories, to optimize the path-integral, we only need to change the
background metric locally. Therefore we can suppress the dependence on coupling constants
λ
0
. In two dimensions, the metric can be chosen to be conformally flat,
ds
2
= e
2φ(x,z)
(dz
2
+ dx
2
). (3.1)
1
Intuitively we can think that we have written the total path-integral as a product of transfer matrices
and each of these tensors represents these transfer matrices.
– 5 –
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