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首页字符演算揭示张量模型中的非平凡相关器结构
本文主要探讨了在张量模型中,特别是在矩阵模型和张量模型之间技术手段的相似性,通过利用Hurwitz字符演算进行计算。字符演算是一种数学工具,它在处理对称群(如Symmetric Group Sm)中的特征问题时表现出高效性。在[20]的工作中,研究者首次展示了高斯相关器在张量模型中展现出非平凡的结构,这对于理解这些复杂模型中的物理现象至关重要。 张量模型中的关键概念是“高斯相关器”,它们是模型中基本的量子态之间的关联度,对于理论的对称性和稳定性有着重要影响。作者提出,对于等级为r的张量,其2m次高斯相关器可以通过将维度以特定的方式线性组合表示,这个组合由大小为m的杨氏图(Young diagrams)给出。杨氏图是一种用于描述多体量子系统的图形工具,它反映了量子数的分布和对称性。 系数部分源自对称组Sm的特征,这些特征依赖于所选择的具体相关器类型以及模型的内在对称性。这意味着不同类型的高斯相关器可能对应不同的特征值分布,这在理论预测和模型分析中具有显著的意义。 例如,在亚里士多德张量模型中,作者给出了一个具体的例子,即三维的三线性组合,用以表述相关器。这种表达方式简洁明了,有助于直观地理解和处理复杂的张量模型计算。 这篇文章通过将字符演算应用于张量模型中的相关器计算,不仅揭示了模型间的计算技术共享,还提供了一种新的视角来理解张量模型中复杂的量子态关联。这对于推进张量模型的研究、探索新的物理现象以及可能的数学交叉领域都有着重要的推动作用。
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Physics Letters B 774 (2017) 210–216
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Correlators in tensor models from character calculus
A. Mironov
a,b,c,∗
, A. Morozov
b,c
a
Lebedev Physics Institute, Moscow 119991, Russia
b
ITEP, Moscow 117218, Russia
c
Institute for Information Transmission Problems, Moscow 127994, Russia
a r t i c l e i n f o a b s t r a c t
Article history:
Received
15 June 2017
Received
in revised form 18 September
2017
Accepted
21 September 2017
Available
online 25 September 2017
Editor:
M. Cveti
ˇ
c
We explain how the calculations of [20], which provided the first evidence for non-trivial structures of
Gaussian correlators in tensor models, are efficiently performed with the help of the (Hurwitz) character
calculus. This emphasizes a close similarity between technical methods in matrix and tensor models
and supports a hope to understand the emerging structures in very similar terms. We claim that the
2m-fold Gaussian correlators of rank r tensors are given by r-linear combinations of dimensions with
the Young diagrams of size m. The coefficients are made from the characters of the symmetric group
S
m
and their exact form depends on the choice of the correlator and on the symmetries of the model.
As the simplest application of this new knowledge, we provide simple expressions for correlators in the
Aristotelian tensor model as tri-linear combinations of dimensions.
© 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
Emerging interest [1–20] to tensor models [21] allows one to begin their systematic study. In the framework of non-linear algebra
[22], one does not expect any essential difference between the tensor and matrix calculi, and the only difference is that the latter is well
developed, while the former one, not. Within the systematic approach, the development should proceed in steps, and the first step is
evaluating the Gaussian correlators [23–25] targeted at finding the underlying structures and their adequate analytic description. In [20],
we demonstrated that the structures are indeed present and, non-surprisingly, similar to those in matrix models. To reveal them in full
generality and beauty, one, however, needs to evaluate a lot of quantities, and thus needs an efficient technique for this. The goal of this
letter is to claim that the most effective calculus of this kind based on the character expansions [26] and Hurwitz theory a la [27] is
directly extended from matrix models case [28–30] to the tensor case. A very similar observation is also made in a very recent paper
[31] (see also the latest development in [32]). In the present letter, we show how the complicated expressions from [20] are drastically
simplified by use of the character/Hurwitz calculus.
As
explained in big detail in [20] and [30], the simplest for the Gaussian calculus is the rectangular complex matrix model (RCM)
[33–35], its tensor liftings are now called rainbow models [8]. The field in this model is the N
1
× N
2
matrix M, and the correlators are
labeled by Young diagrams ={m
1
≥ m
2
≥ ...≥ m
l
> 0}:
O
=
l
p=1
(Tr M
¯
M)
m
p
(1)
where
¯
M = M
†
is Hermitian conjugate, i.e. an N
2
× N
1
matrix, while the averages are defined as the 2N
1
N
2
-fold integrals
...
=
e
−Tr M
¯
M
d
2
M, with d
2
M =
N
1
a=1
N
2
α=1
d
2
M
aα
(2)
*
Corresponding author.
E-mail
addresses: mironov@lpi.ru, mironov@itep.ru (A. Mironov), morozov@itep.ru (A. Morozov).
https://doi.org/10.1016/j.physletb.2017.09.063
0370-2693/
© 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.
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