Physics Letters B 797 (2019) 134866
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Field equations for the simplest multi-particle higher-spin systems
I.S. Degtev, M.A. Vasiliev
∗
I.E. Tamm Department of Theoretical Physics, Lebedev Physical Institute, Leninsky prospect 53, 119991, Moscow, Russia
a r t i c l e i n f o a b s t r a c t
Article history:
Received
12 June 2019
Received
in revised form 3 August 2019
Accepted
13 August 2019
Available
online 19 August 2019
Editor:
M. Cveti
ˇ
c
We derive the product law for the simplest multi-particle higher-spin algebra M
2
(A) and its factor-
algebra
M
2
(A) with rank-one fields factored out. Equations of motion for the systems resulting from
these algebras are analysed. We conclude that the equations for M
2
(A) describe the conformal off-shell
higher-spin system with the rank-two fields representing the off-shell degrees of freedom of the originally
massless system. On the other hand the equations resulting from M
2
(A) describe the infinite system of
conserved currents unrelated to massless fields.
© 2019 The Authors. 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
One of the major problems of the theory of fundamental inter-
actions
is to understand relation between String Theory [1], that
contains infinite towers of massive higher-spin (HS) fields, and HS
gauge theories in which all HS fields are massless and which ex-
hibit
infinite-dimensional HS symmetries (for review see, e.g., [2]).
Arguments that String Theory possesses higher symmetries in the
high-energy limit were given long ago in [3,4]. This suggests that
String Theory should be related to HS theory. For related discus-
sion
see, e.g., [5,6].
More
recently, further proposals on the relation between HS
gauge theories and String Theory were put forward in [7,8]. In
[9]it was conjectured that there exists a broad class of HS gauge
theories based on the HS algebras and their further multi-particle
extensions. In [9]it was argued that dynamics based on certain
algebras has no room for usual massless fields while other mod-
els
properly describe HS gauge fields. In this paper, we consider in
some detail how this can be seen at the level of field equations
associated with one or another algebra in the framework of the
simplest multi-particle extension proposed originally in [10].
Specifically,
we consider two versions of the two-particle alge-
bra
and discuss the form of the related field equations. The cor-
responding
algebras denoted M
2
and M
2
, respectively, are factor-
algebras
of the simplest multi-particle algebra of [10]. To these
algebras we associate two HS systems and derive their equations of
motions. It will be shown that the M
2
system properly describes
massless HS fields as well as two-particle fields that can be in-
*
Corresponding author.
E-mail
addresses: degtevis@gmail.com (I.S. Degtev), vasiliev@lpi.ru
(M.A. Vasiliev).
terpreted as conserved currents. In particular, it will be explained
how the M
2
system is related to the system of HS currents built
from products of HS fields according to [11,12]. On the other hand,
it will be explicitly shown that, in agreement with the general
group-theoretical argument of [9], the M
2
system, that can be un-
derstood
as resulting from the oscillator (Weyl) algebra with two
sets of oscillators, cannot describe massless fields.
The
rest of the paper is organized as follows. In Section 2 the
construction of the multi-particle algebras is recalled. In Sections 3
and
4 we analyse field equations of the M
2
and M
2
systems, re-
spectively.
Brief conclusions are in Section 5. Appendix contains
some details of the derivation of the relevant multi-particle alge-
bras.
2. Multi-particle algebras
First, we recall relevant elements of the algebraic construction
of [10]. Let A be some associative algebra A with basis elements
t
i
, the product law and structure coefficients
t
i
t
j
= f
k
ij
t
k
(2.1)
obeying associativity condition
(t
i
t
j
)t
k
= t
i
(t
j
t
k
) ∈ A, t
i
, t
j
, t
k
∈ A . (2.2)
A is also assumed to be unital with the unit element e
obeying
e
t = t e
= t, ∀t ∈ A . (2.3)
For any such A it is possible to build an associative multi-
particle
algebra M( A) having the meaning of the universal en-
veloping
of A (more precisely, of the Lie algebra associated
with A). As a linear space
https://doi.org/10.1016/j.physletb.2019.134866
0370-2693/
© 2019 The Authors. 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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