超微粒理论：弦驱动力下的世界线新视角

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超微粒理论是一种创新的物理学理论，它在古典层面上表现为一种世界线理论，具备了常规的重新参数化不变性，同时引入了两个独特的特性。首先，该理论源于弦理论的研究，尤其是在紧凑目标上的弦论背景，它试图在非相互作用的层次上理解弦理论中的粒子行为，或者作为Born几何的一种粒子模型的实现。在这个模型中，关键的一点是添加了一个额外的局部对称性，从弦论的角度看，这是世界表面非同构不变性的完备体现；从粒子世界的视角，它反映了额外的局部约束。第二个独特之处在于超微粒相空间上存在非平凡的辛形式，这一概念受到了弦理论的启发，特别是由Laurent Freidel、Robert G. Leigh和Djordje Minic在一系列研究中提出的。他们在论文《闭弦零模式的非交换性》（Phys. Rev. D 96, 066003 (2017)）和《封闭弦理论的内在非交换性》（J. High Energy Phys. 09(2017)060）中探讨了这一现象。这些辛形式的存在表明超微粒的运动在Born几何的时空中是非交换的，类似于弦论中的T对偶性，它与不同的时空选择密切相关。超微粒理论的一个核心议题是其因果性和统一性。在量子层面上，理论的定义和发展旨在探讨粒子行为的量子特性，以及如何将这些特性与经典原理结合起来。超微粒的传播不仅受到传统时空结构的影响，还可能涉及更为复杂的几何和代数结构，这为量子引力和宇宙学提供了一个全新的视角。超微粒理论代表了一种前沿的物理研究方向，它不仅拓展了我们对基本粒子的理解，还挑战了传统的时空观念，展示了弦理论和量子力学之间潜在的深层联系。通过深入研究超微粒理论，科学家们有望揭示新的物理定律，甚至可能引领未来的理论物理学走向新的突破。

we introduce a second constraint

D ¼ p

− 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,

and setting them to zero on quantum states gives the on-

shell conditions for particle states

whose oscillator levels

are associated with values of m

and μ. Thus, in the string

theory analogue, the metaparticle theory corresponds (for

specific values of m

, 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

∧ δx

þ δ

∧ δ

þ πα

δp

∧ δ

: 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

, leads to noncommutativity of x

and

, 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. 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,

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.

Generally, for a function f on phase space, we

define the corresponding Hamiltonian vector field

via

−ωðξ

; ·Þ¼δf. The Poisson bracket of functions f and g is

given by

ff; gg ≡ ξ

ðgÞ¼δgðξ

Þ¼ωðξ

; ξ

Þ: ð8Þ

The symplectic structure (7), therefore, implies the follow-

ing nontrivial equal-time brackets:

g¼δ

; f

;

g¼δ

; f

g¼πα

: ð9Þ

As mentioned above, we see that the effect of the α

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

. 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

coordina-

tize spacetime, with

representing additional quantum

numbers. It should be clear though that there is nothing

special about the x

; 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

dτ½p ·

x −

x ·

p þ πα

p ·

p − eH −

eD: ð10Þ

Its variation leads to the equations of motion of the form

μν

þπα

; H ¼ 0;

¼ 0 ð11Þ

μν

− πα

; D ¼ 0;

¼ 0 ð12Þ

Thus, we see that in the case of a free classical

metaparticle, the effects of the α

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

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.

H and D are associated with world-sheet time and space

reparametrizations, respectively.

By allowing (the eigenvalues of)

p to be nonzero, we are

effectively compactifying the target space. In standard notation,

we have α

=2 ¼ N þ

N − 2 and μα

¼ N −

N. Thus, in the

string interpretation, m

, μ are quantized in units of the string

length, although here we simply take them as continuous

parameters.

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].

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

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