伽利略极限下的杨-米尔斯理论新发现：无限维对称性与GCFT示例

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伽利略·杨·米尔斯理论是研究非相对论极限下量子场论的一个重要分支，它在开放存取文献JHEP04(2016)051中被深入探讨。该理论起源于伽利略变换在电动力学中的应用，但在这里扩展到了更广泛的杨米尔斯理论框架。作者们首先专注于SU(2)理论，这是一种基本的规范群，之后通过推广到SU(N)形式，使理论表达和分析过程更为系统化。在伽利略限下，他们发现了一系列前所未有的现象：不同的极限条件导致了伽利略杨米尔斯理论的不同领域，每个领域都揭示出了一种全新的对称性——D=4维度下的无限Galilean保形对称性。这种对称性在四维空间时间中体现了Galilean不变性，对于理解高维相互作用的场论具有重要意义，特别是对于那些非相对论的、保持Galilean不变性的场论（简称GCFT）来说，这标志着D大于2的理论模型的一个突破。研究者们通过细致的分析，展示了这些理论如何在非相对论环境下维持运动方程的对称性，并且它们是如何在不同极限情况下展现出不同的行为。他们的工作不仅扩展了我们对杨米尔斯理论基础的理解，也为探索非相对论物理的新型对称性和可能的新理论模型提供了新的途径。这篇论文的重要性在于它不仅深化了我们对经典物理与量子场论交界处的理论结构的认识，而且可能引领未来的研究方向，特别是在寻找超越相对论的物理现象的新方法上。通过Galilean Yang-Mills理论的研究，科学家们得以进一步检验物理定律在极端速度或低能量条件下的适用性，这对于理论物理学的发展具有深远的影响。

JHEP04(2016)051

2.1 The inﬁnite Galilean conformal symmetry

The group of conformal transformations of the D dimensional Minkowski space R

D−1,1

SO(D, 2). The most obvious way the group of Galilean transformations is obtained is by a

Inonu-Wigner contraction of this group. A more physical space-time interpretation of this

procedure can be gained by noticing that the generators of the original conformal group

can be represented as vector ﬁelds f

(x)∂

on R

D−1,1

. As is evident, the process of going

to the Galilean framework involves breaking of explicit Lorentz covariance in the following

space-time contraction:

→ x

, t → t,  → 0. (2.1)

This scaling of spatial coordinates means including only slow observers as v

∼

→ v

in units of speed of light (c = 1), thus invoking the principle of Galilean relativity. Let’s

describe how the space-time contraction works for vector ﬁelds generating transformations

through an example of the boost generator. The Lorentz boost generator changes as

= t∂

+ x

∂

7→ 

−1

t∂

+ x

∂

under the scaling (2.1). In order to extract the ﬁnite part of it, we deﬁne the Galilean boost

multiplying this by  and the taking the appropriate limit. This results in B

= t∂

. This

algorithm of ‘Galileanization’ can be carried out for all the generators (Poincare, dilatation

and special conformal). As is evident from the example of boost generator, the vector ﬁeld

form of the generators modify and hence do their Lie brackets, resulting a new Lie algebra,

diﬀerent from so(D, 2), which we name as ﬁnite Galilean conformal algebra (f-GCA). A

basis for this algebra is spanned by the vector ﬁelds:

(n)

= −t

n+1

∂

− (n + 1)t

∂

(n)

= t

n+1

∂

for n = 0, ±1 and J

= x

∂

. (2.2)

A more familiar identiﬁcation is L

(−1,0,1)

= H, D, K and M

(−1,0,1)

= P

, B

, K

where H, D

and K are respectively the Galilean Hamiltonian, dilatation and (SO(D −1)-scalar) special

conformal transformation. On the other hand P

, B

and K

represent momentum, Galilean

boost and (SO(D − 1)-vector) special conformal transformation. J

, as usual, generates

homogeneous SO(D − 1) rotations.

Working out the Lie-brackets of the vector ﬁelds (2.2) we can write the full algebra of

f-GCA as

(n)

, L

(m)

] = (n−m)L

(n+m)

, [L

(n)

, M

(m)

]=(n−m)M

(n+m)

, [M

(n)

, M

(m)

] = 0 (2.3)

, J

]=δ

k[i

j]l

− δ

l[i

j]k

, [L

(n)

, J

] = 0, [M

(n)

, J

] = M

(n)

j]i

with n, m = 0, ±1. One very interesting observation of [35] is that the algebra (2.3) closes

even if we let the index n of (2.2) run over all integers. This inﬁnitely enhanced Lie

algebra will be referred to as GCA from now on.

The embedding of f-GCA inside GCA is

An even larger inﬁnite algebra can be obtained if we give a lift to the rotation generators

(n)

= t

∂

We shall however choose not to work with this larger algebra as it does not turn out to leave theories under

consideration invariant. As of now, we don’t understand the reason behind this.

– 5 –

JHEP04(2016)051

therefore similar to that of SL(2, R) in Witt algebra (algebra of smooth diﬀeomorphisms

of S

). Thus there seems to be an inﬁnite enhancement of symmetries when one looks at

non-relativistic limits of conformal ﬁeld theories in any dimension. This claim obviously

needs to be justiﬁed by looking at examples. A partial realisation was achieved in [7] when

one considered non-relativistic hydrodynamics. Here one found that the Euler and the

Navier-Stokes equations were invariant under arbitrary time dependent boosts, or in the

language of the algebra, invariant under all M

(n)

. But until recently, there was no example

of a theory which realised the full GCA as its symmetry in dimensions D > 2. Galilean

Electrodynamics, as we go on to describe now, was the ﬁrst example of a GCFT in D > 2.

2.2 Scale-spin highest weight representation of GCA

To set the stage, we need to discuss aspects of the representation theory constructed in [5].

We will be interested in the scale-spin highest weight representations, where the states are

labelled by weights under the dilatation and rotation generators as opposed to the scale-

boost representations of [35] which, e.g. are of fundamental relevance in the D = 2 case [8].

For further discussion on this, the reader is referred to [5]. As just stated, we would label

our states by the weights under L

(0)

and J

(0)

|Φi = ∆|Φi, J

|Φi = Σ

|Φi. (2.4)

Then we deﬁne the primary states in a way similar to usual conformal ﬁeld theories by

demanding that the spectrum be bounded from below and hence these primary states are

annihilated by the annihilation operators of the algebra. The primary state conditions are

then given by:

(n)

|Φi

= M

(n)

|Φi

= 0 ∀n > 0. (2.5)

To study the action of the GCA on the operators, we propose a state-operator correspon-

dence, again in close analogy with conformal ﬁeld theories, in order to have a relation

between primary state and the vacuum:

|Φi

= Φ(0, 0)|0i.

The action of f-GCA on primaries is given by:

, Φ(0, 0)] = Σ

Φ(0, 0),

(0)

, Φ(0, 0)

= ∆Φ(0, 0), (2.6a)

(−1)

, Φ(t, x

)

= ∂

Φ(t, x

(−1)

, Φ(t, x

)

= −∂

Φ(t, x

), (2.6b)

(+1)

, Φ(0, 0)

= 0 =

(+1)

, Φ(0, 0)

. (2.6c)

At general space-time points (t, x

), it is straightforward to work out the action of the

generators of the GCA on operators:

Φ(t, x) = UΦ(0, 0)U

−1

with U = e

(tL

(−1)

−x

(−1)

)

(2.7)

For a general GCA element O, we have



O, Φ(t, x

)



= U



−1

OU, Φ(0, 0)



−1

– 6 –

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