![](https://csdnimg.cn/release/download_crawler_static/12384365/bg3.jpg)
we introduce a second constraint
D ¼ p
μ
˜
p
μ
− μ; ð6Þ
along with a second Lagrange multiplier that we call
˜
e.In
the analogous string theory, these two constraints are
associated with world-sheet diffeomorphism invariance,
3
and setting them to zero on quantum states gives the on-
shell conditions for particle states
4
whose oscillator levels
are associated with values of m
2
and μ. Thus, in the string
theory analogue, the metaparticle theory corresponds (for
specific values of m
2
, μ) to a formulation of the dynamics of
particle states at a fixed level. Later in the paper, we will
address positivity requirements on e and
˜
e. We will also
discuss later the symmetries of the model; here we simply
note that the H-constraint is invariant under Oð2; 2d − 2Þ,
while the D-constraint breaks that to a subgroup which
includes an apparent doubling of the Lorentz group
Oð1;d− 1Þ.
The second ingredient in the specification of the meta-
particle is the symplectic structure, which we take to be
ω ¼ δp
μ
∧ δx
μ
þ δ
˜
p
μ
∧ δ
˜
x
μ
þ πα
0
δp
μ
∧ δ
˜
p
μ
: ð7Þ
This form of the symplectic structure is again motivated by
that of the zero modes of the Polyakov string on compact
spacetime [13,14]. The third term, which depends on an
additional parameter α
0
, leads to noncommutativity of x
μ
and
˜
x
μ
, and so the phase space coordinates that we are using
here are not quite Darboux coordinates. One recovers the
Darboux parametrization in the limit α
0
→ 0. It may seem
artificial to not simply diagonalize the symplectic form, but
we choose not to do so as, in the context of string theory x,
˜
x
are preferred coordinates, and more generally we expect
that the introduction of interactions will single these out as
the most natural. The third term in the symplectic form has
interesting consequences for the dynamics of metapar-
ticles.
5
Generally, for a function f on phase space, we
define the corresponding Hamiltonian vector field
6
ξ
f
via
−ωðξ
f
; ·Þ¼δf. The Poisson bracket of functions f and g is
given by
ff; gg ≡ ξ
f
ðgÞ¼δgðξ
f
Þ¼ωðξ
f
; ξ
g
Þ: ð8Þ
The symplectic structure (7), therefore, implies the follow-
ing nontrivial equal-time brackets:
fp
μ
;x
ν
g¼δ
ν
μ
; f
˜
p
μ
;
˜
x
ν
g¼δ
ν
μ
; f
˜
x
μ
;x
ν
g¼πα
0
δ
ν
μ
: ð9Þ
As mentioned above, we see that the effect of the α
0
term
is to render the coordinates (x,
˜
x) noncommutative.
Consequently, if we were to reduce from phase space to
the Born geometry x,
˜
x, the resulting theory would be
nonlocal on the world line. We will not refer to this further
in the present paper, in favor of discussing the theory in a
particular Lagrangian subspace, which we will take to be
coordinatized by x
μ
,
˜
p
μ
. In the quantum theory, we specify
this as a choice of transition amplitude, related to world
lines with fixed x,
˜
p at its endpoints. As we will see, this
has an interesting interpretation in which the x
μ
coordina-
tize spacetime, with
˜
p
μ
representing additional quantum
numbers. It should be clear though that there is nothing
special about the x
μ
; we could have taken any linear
subspace, such as
˜
x, p. Generally, other polarizations
can be obtained from our analysis by generalized
Fourier transforms and will be considered elsewhere.
Let us begin by exploring some classical aspects of the
metaparticle. The detailed form of the action depends on
the above-mentioned choice of boundary conditions, which
specifies a choice of presymplectic 1-form Θ, with ω ¼ δΘ.
In the present case, the metaparticle action is given by
S ¼
Z
dτ½p ·
_
x −
˜
x ·
_
˜
p þ πα
0
p ·
_
˜
p − eH −
˜
eD: ð10Þ
Its variation leads to the equations of motion of the form
eh
μν
p
ν
þ
˜
e
˜
p
μ
¼
_
x
μ
þπα
0
_
˜
p
μ
; H ¼ 0;
_
p
μ
¼ 0 ð11Þ
eh
μν
˜
p
ν
þ
˜
ep
μ
¼
_
˜
x
μ
− πα
0
_
p
μ
; D ¼ 0;
_
˜
p
μ
¼ 0 ð12Þ
Thus, we see that in the case of a free classical
metaparticle, the effects of the α
0
term drop out if we
impose the (x,
˜
x) equations of motion, that force the
momenta to be constant. Thus, were it not for the D gauge
constraint, the classical motion would apparently be that of
a particle in a doubled spacetime. The α
0
term will however
have a nontrivial effect in the quantum theory, especially in
the presence of interactions. One of the reasons for the need
for an extra constraint is that the doubling of position
variables also implies a doubling of their time components.
We therefore need two constraints to saturate each of the
time directions and ensure that the metaparticle dynamics is
causal. We will provide much more detail on this point in
what follows.
3
H and D are associated with world-sheet time and space
reparametrizations, respectively.
4
By allowing (the eigenvalues of)
˜
p to be nonzero, we are
effectively compactifying the target space. In standard notation,
we have α
0
m
2
=2 ¼ N þ
˜
N − 2 and μα
0
¼ N −
˜
N. Thus, in the
string interpretation, m
2
, μ are quantized in units of the string
length, although here we simply take them as continuous
parameters.
5
It is useful to note that the Poisson brackets derived from (7)
are reminiscent of the Dirac brackets found for charged particles
in a magnetic field, reduced to the lowest Landau level. This
observation is useful in understanding the canonical quantization
of the system [12].
6
When ω is closed and nondegenerate at least, this Hamiltonian
vector field is unique.
THEORY OF METAPARTICLES PHYS. REV. D 99, 066011 (2019)
066011-3