Some Notes Concerning the Dynamics of Noncommutative Lumps Corresponding to Nontrivial Vacua in Noncommutative Yang--Mills Models which are perturbative branches of M(atrix) Theory

Abstract

We consider a pair of noncommutative lumps in the noncommutative Yang--Mills/M(atrix) model. In the case when the lumps are separated by a finite distance their ``polarisations'' do not belong to orthogonal subspaces of the Hilbert space. In this case the interaction between lumps is nontrivial. We analyse the dynamics arisen due to this interaction in both naive approach of rigid lumps and exactly as described by the underlying gauge model. It appears that the exact description is given in terms of finite matrix models/multidimensional mechanics whose dimensionality depends on the initial conditions.

Full text

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SOME NOTES CONCERNING THE DYNAMICS OF
NONCOMMUTATIVE LUMPS CORRESPONDING TO
NONTRIVIAL VACUA IN THE NONCOMMUTATIVE
YANG–MILLS MODELS WHICH ARE PERTURBATIVE
BRANCHES OF M(ATRIX) THEORY
CORNELIU SOCHICHIU
Abstract. We consider a pair of noncommutative lumps in the noncommuta-
tive Yang–Mills/M(atrix) model. In the case when the lumps are separated by
a finite distance their “polarisations” do not belong to orthogonal subspaces
of the Hilbert space. In this case the interaction between lumps is nontrivial.
We analyse the dynamics arisen due to this interaction in both naive approach
of rigid lumps and exactly as described by the underlying gauge model. It
appears that the exact description is given in terms of finite matrix mod-
els/multidimensional mechanics whose dimensionality depends on the initial
conditions.
1. Introduction
Recent progress in theories over noncommutative spaces (for a review see e.g.
[1]–[4] and references therein), is stimulated by their importance for the nonpertur-
bative dynamics of string theory [5]–[8].
The noncommutative models share some common features with their commuta-
tive counterparts, however, there is a striking dissimilarity between them in some
other aspects. One particular feature of noncommutative field theories discovered
recently and which attracted a considerable interest is that in noncommutative
models there exists a kind of localised solutions nonexistent in the models on com-
mutative spaces. Although they are different from what is a soliton in usual sense,
these solutions are conventionally called “noncommutative solitons”. In actual work
we consider a subclass of such configurations. As it appears that the “noncommu-
tative solitons” in actual work even do not carry energy (at rest) more adequate
would be the term of “vacuum” or “lump” solution, throughout this paper we will
keep the last name for them.
Noncommutative lumps, in a scalar model with a potential having nontrivial
local minima were first discovered in [9], in the limit of strong noncommutativity.
These solutions were interpreted as condensed lower dimensional branes living on
a noncommutative brane [10, 11]. They were further generalised to the case of
a mild noncommutativity by allowing the presence of the nontrivial gauge field
backgrounds [12]–[15]. This solutions correspond to nontrivial gauge field fluxes
[16]–[18].The particular property of lump solutions we are considering in the
actual paper is that they are “made” purely of the gauge fields. However using the
equivalence between different noncommutative Yang–Mills–Higgs models [19, 20],
these configurations can be mapped into noncommutative solitons in the sense of
Ref. [14] or others.
The general multi-lump solutions look like sums of projectors to mutually or-
thogonal finite-dimensional subspaces of the Hilbert space. If subspaces are not
Work supported by RFBR grant #99-01-00190, INTAS grants #1A-262 and #99 0590, Scien-
tific School support grant # 00-15-96046 and Nato fellowship program.
1

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2CORNELIU SOCHICHIU
orthogonal the configuration fails to be a static solution and lumps start to inter-
act.
An approach to describe interacting lumps were proposed in [21, 22] by using
the substitution of the configuration of shifted lumps by a close one but being a
static solution. In this approach the interaction of the lumps is described by the
motion in the curved moduli space of static solutions. This approach, however,
would be valid only provided that the motion is confined to the moduli space of the
static solution which requires it to be stable. There are, however, indications that
the noncommutative lumps are not stable dynamically [23] which leads also to the
instability of the motion around the moduli space.
Our approach is free of these drawbacks since we do not make any assumptions
about the stability. As the analysis shows the dynamics of the system does not look
as a stable one, moreover, it appears to be stochastic! The regular motion occurs
only when the distance between lumps is exactly
that for some natural initial conditions the dynamics of noncommutative lumps is
described by finite dimensional matrix model.
The plan of the actual paper is as follows. First, we introduce the reader to
the noncommutative lumps in Yang–Mills–Higgs model. After that we analyse the
lump dynamics in both naive approach when we treat lumps as rigid particles and
neglect the dynamics of the “shapes” of the lumps and in an exact approach when all
possible deformations are taken into account. The comparison reveals unexpected
features in the behaviour of the interacting lumps.
generalise the description to the case of lumps with arbitrary polarisations.
√θln2. It is interesting to note
We also give directions to
2. The Model
In this paper we consider the noncommutative gauge model described by the
following action,
?
Here fields Xi, i = 1,...,D are time dependent Hermitian operators, acting on
Hilbert space H which realises a irreducible representation for the one-dimensional
Heisenberg algebra generated by,
S =dttr
?1
2
˙Xi˙Xi+
1
4g2[Xi,Xj]2
?
.
(1)
[x1,x2] = iθ.
(2)
Operators xµsatisfying the algebra (2) are said also to be the coordinates of a
noncommutative two-dimensional plane. In this interpretation the operators of the
Heisenberg algebra H can be represented through ordinary functions given by their
Weyl symbols. The composition rule for the symbols is given by the Moyal or star
product,
f ∗ g(x) = eiθ?µν∂µ∂?
νf(x)g(x?)
???
x?=x,
(3)
where f(x) and g(x) are Weyl symbols of some operators, f ∗ g(x) is the Weyl
symbol of their product and ∂µ,∂?
x?µ.Integration of a Weyl symbol corresponds to 2πθ×trace of the respective
operator, while the partial derivative derivative with respect to xµcorresponds to
the commutator,
µdenotes the derivatives with respect to xµand
∂µf(x) = i(pµ∗ f − f ∗ pµ)(x) = [pµ,f](x),
where pµ is given by pµ = (1/θ)?µνxν. Since there is one-to-one correspondence
between operators and their Weyl symbols we will not distinguish between them,
i.e. keep the same character for both, unless in the danger of confusion.
(4)

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DYNAMICS OF VACUA . ..3
The model (1) corresponds to the Hilbert space (N → ∞ limit) of the BFSS
Matrix Model as well as in different perturbative limits it describes the noncom-
mutative Yang–Mills(–Higgs) model in the temporal gauge A0= 01[24, 19, 20].
Indeed, for equations of motion corresponding to the action (1),
¨ Xi+1
g2[Xi,[Xi,Xj]] = 0,
one may find static classical solution Xi= pi, [24, 19, 20], satisfying,
(5)
[pi,pj] = iθ−1
ij,
(6)
with constant invertible θ−1
condition: from [pi,F] = 0 with all pi, i = 1,...,D it follows that F is a c-number,
F ∼ I.
Expanding fields around this solution, Xi = pi+ Ai, and Weyl ordering op-
erators Ai with respect to xi= θijpj one gets precisely the (D + 1)-dimensional
noncommutative Yang–Mills model for the field given by the Weyl symbol Ai(x).
Getting another solution with a smaller number of independent pi’s, Xα= pα,
α = 1,...,p,
ij. We assume about the solution also the irreducibility
[pα,pβ] = iθ−1
αβ,
(7)
and Xi = const, i = p + 1,p + 2,...,D = 0 one gets as a result the model of
p-dimensional Yang–Mills field interacting with (D − p) scalars.
Having in mind this equivalence, in what follows we will consider two-dimensional
form of this noncommutative model. If one forgets for a while also the issues with
the Gauss law the theory looks like a “simple” noncommutative scalar model in
(2+1) dimensions.
For our purposes it will be convenient to use two-dimensional “complex coordi-
nates” given by oscillator rising and lowering operators2a and ¯ a,
1
√2θ(x1+ ix2),
and the oscillator basis,
a|n? =√n|n − 1?,
As one can see the solution (6) or (7) has divergent traces. Another type of static
solutions one can find in the model (1) is given by a configuration with localised
i.e. lump-like Weyl symbols (in some background pi).3It is given by commutative
matrices of finite ranks [14]. Although these lumps carry no energy, — they are
geometrically nontrivial vacua, we will call them noncommutative lumps due to
their close relation to ones discussed in the literature [9]–[15].
Up to a gauge transformation the N-lump solution is given by,
a =
¯ a =
1
√2θ(x1− ix2),
[a,¯ a] = 1,
(8)
¯ aa|n? = n|n?,
¯ a|n? =√n + 1|n + 1?.
(9)
Xi=
N
?
n=0
ci
n|n??n|,
(10)
where ci
of the rank the Weyl symbol of Xivanishes at infinity as quick as Gaussian factor
nis n-th eigenvalue of the (finite rank) operator Xi. Due to the finiteness
1One should take care that the Gauss law constraints are satisfied as well. We postpone the
discussion on the Gauss law constraints until the subsection 3.4
2For the Weyl symbols we will use later z and ¯ z instead of a and ¯ a to distinguish them from
the Hilbert space operators.
3Fairly speaking these solutions are localised if the fields are treated as scalar ones. Since the
gauge field definition Aα = Xα− pα implies substraction of a linear function pα this type of
solutions correspond to functions with linear growth.

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4CORNELIU SOCHICHIU
times a polynomial. The simplest one-lump solutions can be written in the form,
X(0)
i
= ci|0??0|,
(11)
where cigive the “height” and the “orientation” of the lump, by a proper Lorentz
transformation, Xi→ Λj
isation” corresponds to the oscillator vacuum state |0?.
In the star-product form operator (11) is represented by the Weyl symbol
iXjit can be made of the form ci= cδi1, while its “polar-
Xi(¯ z,z) = 2cie−2|z|2.
The lump shifted along noncommutative plane by a (c-number) vector u is given
by
X(u)
i
= cie−ipµuµ|0??0|eipµuµ= cie−|u|2e¯ au|0??0|e−a¯ u.
Its Weyl symbol, correspondingly, is given by X(u)
lump with constant u is a solution again. When u becomes time-dependent one
can perform the time-dependent gauge transformation to get4,
(12)
i
(z) = 2cie−|z−u|2. The shifted
Xi→ eipµuµXie−ipµuµ,
(13)
which shifts the lump back to the centre, but produce a kinetic term for ∼ ˙ u2/2.
Thus a single noncommutative lump moves freely like a non-relativistic particle. It
is also stable since its energy at rest is zero.
In what follows we are going to analyse the situation when there is a couple of
lumps separated by a distance u.
3. A pair of interacting lumps
As we have shown in the previous section, a single noncommutative lump can be
always rotated to have the polarisation |0? and orientation along Xi. When there
are just two lumps one can choose without loss of generality the configuration to
involve nontrivially only two matrices e.g. X1and X2.
Consider two lumps which are obtained from c|0??0| by shifts along the non-
commutative plane by respectively u(1)and u(2). Since the dynamics of the centre
is free and can be decoupled by a time-dependent gauge transformation similar to
(13), where u = u(1)+ u(2)is the coordinate of the centre.
Thus, the configuration we consider looks like,
X1= c1V PV−1≡ c|−u/2??−u/2|,
X2= c2V−1PV ≡ c|u/2??u/2|
Xi= const,i = 3,...,D,
(14a)
(14b)
(14c)
where we introduced the shorthand notations,
V = e(i/2)pµuµ= e
P = |0??0|.
1
2(a¯ u−¯ au),
(15)
(16)
The quotient c can be absorbed by the rescaling of the coupling and the time,
therefore we can put it generically to unity c = 1.
In what follows also Xii = 3,...,D will enter trivially in the equations, so in
the remaining part of the paper for the simplicity of notations the index i will run
the range i = 1,2. If we were considering more than two noncommutative lumps
we would have to keep more matrices.
4Remember the gauge: A0= 0.

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DYNAMICS OF VACUA . ..5
u
V
Figure 1. The profile of the lump-lump interaction potential in
the naive approach.
3.1. Naive picture: rigid lumps. Consider first a naive approach where we
are dealing with rigid interacting lumps which means that we are neglecting the
deformations in their shapes. In this case the only parameter which is dynamical
is the separation distance u.Although, this approximation sounds reasonable,
later we will consider the exact description which shows that this approach is not
justified. However, we decided to keep this naive analysis for illustrative purposes.
To obtain the action describing the dynamics let us insert the ansatz (14) into
the classical action (1). The computation of derivatives and traces gives for the
following u-dependence of the action,
?
where we restored the explicit θ dependence.
The potential is depicted on the fig.1. According to it sufficiently close lumps
attract while distant ones repel. At the critical distance uc =
stay in unstable equilibrium.
The above conclusions concerning the lump dynamics would be valid, however,
only in the case when one can neglect the involvement of the lump shape in the
dynamics. To evaluate the importance of the shape dynamics one should consider
arbitrary deformations of the shape of lumps and separate them from the motion
of the lump as a whole.
In the next subsection we analyse the dynamics from the point of view of exact
field equations of motion. The lump configuration is taken to be the initial condition
for the field equations. The result we obtain in the next section will invalidate the
results of the actual naive approach, however, the critical distance ucwill correspond
to a special case.
S[u] =dt
?1
2θ|˙ u|2− 2e−|u|2
2θ(1 − e−|u|2
2θ)
?
,
(17)
√2θln2 they will
3.2. Exact description: Lumps at rest. The exact description of the lump
dynamics is given by the field equations of motion for Xi,
¨ Xi+1
g2[Xj,[Xj,Xi]] = 0
corresponding to the action (1), supplied with initial conditions given by the lump
background (14). Since the equations are second order, in addition to this one has
to consider the initial values for the time derivatives of Xi. The simplest choice is
when the configuration at t = 0 is static. Thus, the initial conditions we impose
are as follows,
(18)
X1|t=0= |−u/2??−u/2|,
X2|t=0= |u/2??u/2|,
Considering the lumps in the initial moment as being at rest, produces a con-
siderable simplification to the equations of motion. Indeed, the initial data (14)
˙X1|t=0= 0,
˙X2|t=0= 0.
(19a)
(19b)

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6CORNELIU SOCHICHIU
imply that the operators Xiare nonzero only on the two-dimensional subspace Hu
of the Hilbert space which is the linear span of vectors |u/2? and |−u/2?. Since,
in virtue of equations of motion (18), the second time derivative is proportional to
commutators of Xithen it also vanishes outside the two-dimensional subspace Hu.
Due to zero initial conditions for the first derivatives operators Xiwill remain all
the time in the same two-dimensional subspace of the Hilbert space.
Let us consider only those components of Xi which are nonzero. This reduces
the Hilbert space operators to ones acting on the two-dimensional subspace Huof
the Hilbert space spanned by |±u/2?. Let us introduce an orthonormal basis on
Hu.
The natural orthonormal basis one can build up out of |±u/2?, is given by vectors
|±?, defined as follows (see the Appendix),
|+? ≡
0
2(1 + e−1
?0
2(1 − e−1
The singularity in the |−? in the limit u → 0 appears since in this limit |u/2? and
|−u/2? tends to be parallel and the subspace become one-dimensional.
In this basis our problem is reformulated in terms of the 2×2 matrix model with
equations of motion superficially looking the same as (18),
+1
g2[X(2)
?1
?
?
=
1
?
?
2|u|2)
(|u/2? + |−u/2?),
(20a)
|−? ≡
1
=
1
2|u|2)
(|u/2? − |−u/2?).
(20b)
¨ X(2)
i
k,[X(2)
k,X(2)
i
]] = 0,
(21)
but now X(2)
basis (20) are rewritten as follows,
i
are finite dimensional 2 × 2 matrices. The initial conditions in the
?
?
(It is worthwhile to note that the description in terms of 2×2 matrices is valid
only for the situation where the lumps were initially at rest.
conditions˙Xi|t=0= 0 is an additional specification and it says that also the shapes
of the lumps are not changing at the initial moment. One may try, however relax
the last condition and consider more general initial configurations including lumps
with rising/decreasing height for ˙Xi ?= 0. One can consider more general initial
condition ˙Xi|t=0 ∝ Xi for which the same description in terms of 2×2 matrices
remains valid. Solutions of this type for u = 0 were considered in [25].)
Similar equations as ones given by (21) although in a different context of the
one-dimensional ordinary Yang–Mills model were under study for a long time and
were initiated by [26]–[28]. In the modern context of the application to the finite
N matrix model they appear in [29]–[32]. The system described by such equations
was shown to exhibit a stochastic behaviour. Let us describe it in more details to
the application to the present case.
In order to rewrite the equations (22) in the scalar form let us expand the
matrices Xiin terms of the two-dimensional Pauli matrices σα, α = 1,2,3, and the
two-dimensional unity matrix I2(which in fact is the projector to Hu) satisfying,
[σα,σβ] = i?αβγσγ,
(23)
X(2)
1|t=0=1
2
1 + e−1
−√1 − e−|u|2
1 + e−1
√1 − e−|u|2
2|u|2
−√1 − e−|u|2
1 − e−1
√1 − e−|u|2
1 − e−1
2|u|2
?
?
,
˙X1|t=0= 0;(22a)
X(2)
2|t=0=1
2
2|u|2
2|u|2
,
˙X2|t=0= 0.
(22b)
Beyond this the
[σα,I2] = 0,

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DYNAMICS OF VACUA . ..7
The expansion is as follows,
X1,2= X0
1,2I2+ Xα
1,2σα.
(24)
In terms of this expansion the equations of motion look as follows,
¨ X0
(25a)
1+1
g2(X2
¨ Xα
g2(X2
where X2
˙Xα
(25d)
2|t=0=1
2
1|t=0= −1
2
X2
(25g)
1,2= 0
¨ Xα
2δα
β− Xα
2X2β)Xβ
1= 0 (25b)
2+1
1δα
β− Xα
1X1β)Xβ
2= 0,
(25c)
1,2= Xα
1,2Xα
1,2. For the initial conditions one also has,
1,2= 0,
X0
1|t=0= X0
(25e)
X1
?
1 − e−|u|2,
2|t=0= 0,
2|t=0= e−1
X1
2|t=0=1
2
?
1 − e−|u|2,
(25f)
1|t=0= X2
X3
1|t=0= X3
2|u|2.
(25h)
In particular, the equation (25a) says that the scalar parts of the matrices X1,2
remains constant during the motion (X0
ditions for the “velocities” ˙X0
subjects to more complicate nonlinear dynamics.
Before analysing the solutions for Xα
tion in terms of the lump dynamics over noncommutative space in the star-product
representation.
The noncommutative function which corresponds to a particular solution Xα
will be given by
Xi(t;z, ¯ z) =1
2I(z, ¯ z) + Xα
where Xα
corresponding to the Pauli matrices.
The respective Weyl symbols are computed in the Appendix. They are given by,
1,2(t) = 1/2) provided the zero initial con-
i= 0. At the same time the remaining parts are
1,2, α = 1,2,3 let us consider their interpreta-
i(t)
i(t)σα(z, ¯ z),
(26)
i(t) are the solutions of to (25) and I(z, ¯ z) with σα(x) are the Weyl symbols
σ1(z, ¯ z) =
2
√1 − e−|u|2
2ie−2¯ zz
√1 − e−|u|2
?
?e¯ zu−¯ uz− e−¯ zu+¯ uz?,
?
?e¯ zu−¯ uz+ e−¯ zu+¯ uz?,
2
1 − e−|u|2
−2e−2|z|2−1
1 − e−|u|2
e−2|z−u
2|2− e−2|z+u
2|2?
,
(27a)
σ2(z, ¯ z) =(27b)
σ3(z, ¯ z) = −2e−1
2|u|2
1 − e−|u|2
+
1 − e−|u|2
e−2|z−u
2|2+ e−2|z+u
2|2?
(27c)
e−2¯ zz
I(¯ z,z) = σ0(z, ¯ z) =
?
e−2|z−u
2|2+ e−2|z+u
2|2?
(27d)
2|u|2
?e¯ zu−¯ uz+ e−¯ zu+¯ uz?.
As it can be seen from the eqs. (26), (27) irrelevant to the particular form of the
solution Xα
(of the size of the order of ∼
i(t), at any time fields Xi(z, ¯ z) are nonzero only in the small vicinities
√θ) of of points z = 0 and z = ±u/2. This means

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8CORNELIU SOCHICHIU
that irrespective to the initial distance between the lumps once left with zero initial
velocities they will not try to leave their places, the dynamics instead will concern
only their heights and creation of a “baby-lump” in the middle point between them.
Let us note that this should be surprising as it is in total disagreement with the
naive approach drawn in the previous subsection, since there is no regime when the
lumps would behave like rigid particles.
Let us consider now the the time-dependent functions Xα
The equations (25) are too complicate to find the general solution, however, for
our particular initial data one can use the rich symmetry of the model and find a
simplifying ansatz.
Assuming that the magnitudes of Xα
nonzero times, we can check this assumption later as a consistency condition for
the ansatz, but also prove it independently of the ansatz using conservation laws,
one can split Xα
i(t) in more details.
1and Xα
2are equal X2
1(t) = X2
2(t) also for
1and Xα
Xα
X =1
2into two orthogonal components Xαand Yαas follows,
Xα
Y =1
2(Xα
the equality of the square modules X2
orthogonal. The equations of motion in terms of X and Y read,
¨ Xα= −2
¨Yα= −2
where X2= XαXαand Y2= YαYα. The initial conditions are respectively,
Xα|t=0=1
˙Xα(0) =˙Yα(0) = 0.
(29d)
1= Xα+ Yα,
2= Xα− Yα,
(28a)
2(Xα
1+ Xα
2),
1− Xα
2),
(28b)
1(t) = X2
2(t) implies that X and Y remain
g2Y2Xα,
(29a)
g2X2Yα,
(29b)
2(Xα
1(0) + Xα
2(0)),Yα|t=0=1
2(Xα
1(0) − Xα
2(0)),
(29c)
From eqs. (29) one can see that the directions of Xαand Yαdo not change.
The fact that Xαand Yαare always mutually orthogonal makes the assumption
X2
Splitting the vectors Xαand Yαin the dynamical magnitude and the static
direction, given by the constant unimodular vectors
√
X2|t=0= (0,0,1),
eα
Y2|t=0= (1,0,0),
one has the equations for the magnitudes X and Y ,
¨ X = −2
¨Y = −2
which are supplied by the initial data,
1(t) = X2
2(t) for the ansatz (28) consistent.
eα
X= Xα/
(30a)
Y= Yα/
√
(30b)
g2Y2X,
(31a)
g2X2Y,
(31b)
X|t=0= e−1
2|u|2,Y |t=0= −
?
1 − e−|u|2,
˙X|t=0=˙Yt=0= 0.
(31c)
As we mentioned earlier, the system (31) exhibits a stochastic behaviour which
has been studied both numerically and analytically [26]–[32]. The system is equiv-
alent to one of a two-dimensional particle in the potential U(X,Y ) = X2Y2. The
allowed by energy conservation region of the configuration space is divided into so
called stadium X ∼ Y ? 1 where the motion is almost free and four channels along

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DYNAMICS OF VACUA . ..9
the axes. The motion in channels can be described by the asymptotic formula, in
the limit when one coordinate is much smaller than another, say X ? Y [33],
Y (t) = −A
1
g2
2t2+ W0t + Y0
?
(32a)
X(t) =
2A
Y (t)cos
?
g
?
−A
6t3+W0
2t2Y0t + ϕ0
??
,
(32b)
where
A =V2
0+ g2X2
Y0
0Y2
0
,ϕ0= arccos
?
X2
0Y2
0
V2
0+ g2X2
0Y2
0
(32c)
and
X0= X(t0),
V0=˙X(t0),
Y0= Y (t0),
W0=˙Y (t0),
(32d)
(32e)
t0being the time of entrance into the channel.
In the channel the particle reach the maximal value of Y ∼ W2
it is reflected back to the stadium. The instability arises when the particle passes
through the stadium and enters a new channel. So, generally the motion of the
particle is stochastic. Also there is a discrete set of trajectories which are closed.
Thus, depending on initial conditions the system can move in a regular periodic
way, although this motion is unstable as arbitrary small perturbation can push the
system to the stochastic regime.
The asymptotic formulae (32a), (32b) can provide a reliable description of the
system for a certain period of time for extremal cases when the lump centre sep-
aration distance is either large (q ≡ e−|u|2? 1) or small (?1 − q2? 1). Thus,
solution,
Y (t) =q2?1 − q2
?
(q2(1 − q2)1/2/4g2)t2− (1 − q2)1/2)
?
g
0/A after which
if u → ∞ (q ? 1) then for times less than tstoch= gq−1, one has the asymptotic
4g2
t2−
?
1 − q2,
(33a)
X(t) =
q2(1 − q2)1/2
(33b)
× cos
1
?
q2?1 − q2
4g2
t3−
?
1 − q2t
??
.
In the opposite case when the lumps are close one can again give a reliable descrip-
tion for of the dynamics by the asymptotic formula,
X(t) = −q(1 − q2)
?
4g2
t2+ q,
(34a)
Y (t) =
q(1 − q2)
−(q(1 − q2)/4g2)t2+ qcos
?1
g
?
−q(1 − q2)
4g2
t3+ qt
??
,
(34b)
valid for times up to of order tstoch = g(1 − q2)−1/2after which the system ap-
proaches the stadium where we cannot control it.
There is also one particular separation distance which corresponds to periodic
motion. This happens for the initial conditions X|t=0= −Y |t=0= 1/√2 or u =
√θln2. In this case the motion is periodic and is given by X(t) = Y (t) ≡ f(t)

Page 10

10CORNELIU SOCHICHIU
where for f(t) we have the (implicit) formula,
f(t) :
t =
?f
1/√2
du
?1/4 − u4.
(35)
Now, let us recall that in terms of X(t) and Y (t) the dynamical field Xi(t, ¯ z,z)
describing the lumps takes according to eq. (26) the following form,
(36a)
X1(t, ¯ z,z) =1
2σ0(¯ z,z) + X(t)σ3(¯ z,z) + Y (t)σ1(¯ z,z) =
1 − e−1
1 − e−|u|2
+1 − e−1
1 − e−|u|2
+X(t) − e−1
2|z|2X(t) −
?1 − e−|u|2Y (t)
?1 − e−|u|2Y (t)
e−2|z−u
2|2
2|z|2X(t) +
e−2|z+u
2|2
2|u|2
1 − e−|u|2
e−2|z|2(e¯ zu−z¯ u+ e−¯ zu+z¯ u),
and,
(36b)
X2(t, ¯ z,z) =1
2σ0(¯ z,z) + X(t)σ3(¯ z,z) − Y (t)σ1(¯ z,z) =
1 − e−1
1 − e−|u|2
+1 − e−1
1 − e−|u|2
+X(t) − e−1
2|z|2X(t) +
?1 − e−|u|2Y (t)
?1 − e−|u|2Y (t)
e−2|z−u
2|2
2|z|2X(t) −
e−2|z+u
2|2
2|u|2
1 − e−|u|2
e−2|z|2(e¯ zu−z¯ u+ e−¯ zu+z¯ u)
where the functions σ1,3are given by the eqs. (27). Let us note that the function
σ1(¯ z,z) is localised in the points where the lumps are i.e. at z = ±u/2 while the
function σ3(¯ z,z) is localised at both lump positions as well as at the origin where
is the middle of the lump centres connecting line.
The analysis of the solution (36) reveals that once left at their positions the
lumps will not tend to move away from them but engage in a stochastic change
of their heights as well as creation of a “baby” lump in the middle point between
them which is the origin of the noncommutative plane. This process can be reliably
described for a while of time in the limits when the lumps are placed very close or
very far, each case degenerating to stochastic, although correlated variation in the
heights of the lumps.
3.3. Exact description: Lumps in motion. The difference arising when consid-
ering moving lumps consists in the initial values for the velocities. Since a generic
initial condition for the velocities can complicate the system making it back infinite
dimensional we confine ourselves to such initial configurations which correspond to
rigid motion of the lumps.
Thus, one has to replace the initial values for the velocities by the following,
˙Xi|t=0=∂Xi
∂u
or, explicitly, using (14),
˙X1|t=0= −1
˙X2|t=0= −1
where v = ˙ u(t = 0), and solve the infinite dimensional operator equation (5).
˙ u|t=0+∂Xi
∂¯ u
˙¯ u|t=0,
(37)
4(¯ vu + ¯ uv)X1|t=0+1
4(¯ vu + ¯ uv)X2|t=0−1
2(v¯ aX1+ ¯ vX1a)|t=0,
2(v¯ aX2+ ¯ vX2a)|t=0,
(38a)
(38b)

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DYNAMICS OF VACUA . ..11
Applying the same strategy as in the case of lumps at rest we see that the initial
data are given by operators which are nonzero only in a four-dimensional subspace
Hv
vectors |±u/2? together with other two newcomers ¯ a|±u/2? = 2(∂/∂u|±u/2?). Let
us note that they are all linear independent for u ?= 0. Therefore, the system is
reduced to the four-dimensional matrix model.
The treatment of this model differs from what we had with lumps at rest only
in technical details, therefore we will not discuss it more.
The qualitative picture one has in this situation does not change much in com-
parison to the case of lumps a rest. Just as in the previous case there is a stochastic
dynamics of the heights of the lumps and creation of “baby”-lumps while the centres
of the lumps will keep moving with the constant velocities. Indeed, for accelerating
lump operators Xiare nonzero out of the subspace Hv
not happen.
In general, the solution is given by a linear combination with time dependent
coefficients of functions (51), their first derivatives (∂σα/∂u), (∂σα/∂¯ u) and some
of their second derivatives like (∂2σα/∂u∂¯ u).
uof the infinite-dimensional Hilbert space H, which is spanned by our two old
u, which, as we know, does
3.4. The Gauss Law. Once we want to relate our system to the the Yang–
Mills/BFSS model we have to care about the Gauss law constraint, which is ob-
tained from the variation of the A0 component of the original gauge invariant
noncommutative Yang–Mills or BFSS action. This constraint looks like follows,
L = [Xi,˙Xi] = 0.
(39)
As we discussed at the beginning of this section the equations of motion imply
that the quantity (39) is at least conserved. Indeed, using the equations of motion
one has
˙L = i[Xi,¨ Xi] = 0.
(40)
Therefore to get a self-consistent solution for the Yang–Mills/M(atrix) theory
one has to verify that L|t=0= 0. For zero velocity initial conditions this is implied
automatically, while for the moving lumps one has,
??¯ u
where “h.c.” stands for the Hermitian conjugate.
The equation (41) requires velocities v and ¯ v to vanish. However, the violation of
the Gauss law for nonzero velocities may be interpreted as the presence of nontrivial
electric charge. Indeed, in the presence of external sources the Gauss law becomes,
L = i[Xi,˙Xi]|t=0= iv
2− ¯ a
?
X1+
?¯ u
2+ ¯ a
?
X2
?
+ h.c.,
(41)
L?= i[Xi,˙Xi] + ρ = 0,
(42)
where ρ is some electric charge density which appears in the action as a term
∆Scharge=?dp+1xρX0and which is chosen to cancel (41) exactly. (Here we are
As a result we have that the Gauss low is satisfied automatically in the case of the
lumps at rest, while moving lumps generate some background charge distribution.
not going to analyse in which conditions such charge density can be created.)
3.5. More dimensions. One can do analogous analysis in more than 2+1 dimen-
sions.
The only difference which appear in p+1 dimensions is that one has to compute
the Weyl symbols of sigma matrices with respect to a different background e.g. one

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12CORNELIU SOCHICHIU
given by (6). As a result one has equations similar to (27),
2
?
σ2(x) = −
1 − e−1
σ3(x) = −
1 − e−1
+
1 − e−1
I(x) = σ0(x) =
1 − e−1
−4e−x·G·x−1
1 − e−1
where we introduced the notation, u × x = θµνuµxν, µ,ν = 1,...,p, and squares
are computed with the metric G = +√−θ−2. In the basis for which the non-
commutativity matrix θµνtakes the canonical form,


the metric G is diagonal,


...
where
?0
Then the solution is given by equation similar to (26) but with I(x) and σα(x)
instead of respective two-dimensional symbols.
σ1(x) =
1 − e−1
4ie−x·G·x
?
2u·G·u
?
e−(x−u
2)·G·(x−u
2)− e−(x+u
2)·G·(x+u
2)?
,
(43a)
2u·G·usinu × x,
2e−u·G·u
(43b)
2u·G·u
2e−x·G·x
?
e−(x−u
2)·G·(x−u
2)+ e−(x+u
2)·G·(x+u
2)?
(43c)
2u·G·ucosu × x,
2
2u·G·u
4u·G·u
?
e−(x−u
2)·G·(x−u
2)+ e−(x+u
2)·G·(x+u
2)?
(43d)
2u·G·u
cosu × x,
θµν=



θ(1)iσ2
0
0
...
00
0
...
...
...
...
θ(2)iσ2
0
...
θ(3)iσ2
...




,


(44)
Gµν=




θ−1
(1)I2
0
0
00
0
...
...
...
...
θ−1
(2)I2
0
...
θ−1
(3)I2
...


?


,
(45)
iσ2=
1
0
−1
?
, and I2=
?10
10
.
(46)
4. Discussions and Conclusions
In this paper we considered the dynamics of interacting noncommutative lumps.
The naive approach for the dynamics is obtained when one considers the motion
of the lumps as rigid structures and not their deformations. The action in this
approach is given by initial classical action of the noncommutative model computed
when all “degrees of freedom” except the positions of the lumps are frozen. In this
approximation one obtains that the lump pair dynamics is described by a cup-
shaped potential having minimum at the origin, Gaussian decay at the infinity and
an unstable equilibrium at the distance
The exact analysis of the interacting lump dynamics in the the framework of
the original noncommutative theory, however, refute above approximation since it
appears that in fact it is the shape which is affected by the dynamics, while the
motion of the centres of the lumps is not.
It is interesting to note that the problem formulated in noncommutative Yang–
Mills model is reduced to one in finite dimensional matrix model. Thus, in the case
√θ ln2.

Page 13

DYNAMICS OF VACUA . ..13
of two lumps starting at the rest the exact description reduces to a 2 × 2 matrix
model. In particular we have that the U(1) part of these model has trivial dynamics,
while the remaining SU(N) parts generally exhibits stochastic behaviour.
The property of this dynamics that it does not affect the motion along the line
connecting the lumps appears to be anti-intuitive to what one could expect from
interaction of (quasi)particle objects. Let us note that an analogous situation can
be met in the analysis of the vortex interaction in the applications to solid state
physics, [34, 35, 36], where a behaviour similar to one of noncommutative lumps
was observed long ago.
It seems that these results can be easily generalised to the case of lumps with
arbitrary mutual Hilbert space polarisations not related to the shifts along noncom-
mutative space. The dynamics of such lumps or branes does not differ qualitatively
from the case shifted ones, however, in this case we do not have simple physical
picture we can observe. However, from the point of view of mathematical complete-
ness this would be worthy to be considered, and probably will be done in future
research.
It seems that the interpretation in terms of branes when the heights of the lumps
have the meaning of coordinates of the 0-brane in the direction transversal to the
noncommutative brane is the most natural. In this context it appears that the dy-
namics of interacting branes affects only the motion in the transversal directions in
which D0-branes are stochastically “bouncing” around the noncommutative brane.
Translating the above said about lumps to the branes, we have learnt that the
dynamics of two interacting 0-branes is described by U(2) M(atrix) model in the
case when the branes do not move or do not change their polarisations. If the
branes are affected by motion one needs a matrix model of higher dimension to
describe it.
So far we considered only the pair interaction of the lumps. It also would be
of interest to extend the analysis of the actual paper to a greater number of the
lumps, eventually to consider the gas of lumps.
Acknowledgements. I am grateful to Th. Tomaras, E. Kiritsis, K. Anagnos-
topoulos, Nikos Tetradis and Ciprian Acatrinei for the friendly atmosphere and
hospitality during my stay in Crete where this work has been done as well as to
useful discussions and the interest paid to my work. I would like to acknowledge the
discussions on the vortex interactions with Th. Tomaras. I benefited from useful
discussions at QFT dept of Steklov Mathematical Institute in Moscow. Discussion
with I.Ya. Aref’eva and P.B. Medvedev improved my understanding the dynamics
of the SU(2) matrix model.
Appendix A. Useful formula connecting the two dimensional
representation with other representations
To improve the understanding of the paper we summarise here the formula con-
necting the three main representations of the objects used in this paper, Hilbert
space operator, noncommutative functions (Weyl symbols) and two-dimensional
matrices.
The two-dimensional space Hufor u ?= 0 is the span of the two vectors |u/2? =
e−1
state, to which the lump operators X1and X2project to.
The vectors |±u/2? have unity modules but are not orthogonal. One can eas-
ily construct an orthonormal basis consisting of vectors {|+?,|−?} given by eqs.
(20a,20b) (see picture 2).
8|u|2e−1
2¯ au|0? and |−u/2? = e−1
8|u|2e
1
2¯ au|0?, where |0? is the oscillator vacuum

Page 14

14CORNELIU SOCHICHIU
|−>
|+>
|−u/2>
|u/2>
Figure 2. The orthogonal vectors |±? constructed from unimod-
ular but nonorthogonal |±u/2?.
σ0(¯ z,z)
σ1(¯ z,z)
σ2(¯ z,z)
σ3(¯ z,z)
Figure 3. Plots of the profiles of the functions σ0,1,2,3(¯ z,z).
An arbitrary Hermitian operator acting in this two-dimensional subspace can be
expanded in terms of ordinary Pauli matrices and unity matrix,
?1
as follows,
σ0≡ I2=
0
10
?
,σ1=
?01
01
?
,σ2=
?0
−i
0i
?
,σ3=
?10
0
−1
?
,
(47)
X = X0+ Xασα,
(48)
where Xαare computed as,
Xα=1
2trXσα.
(49)
From the other hand, as operators over the Hilbert space the two-dimensional
unity matrix5and Pauli matrices can be expressed as noncommutative functions
through their Weyl symbols.
5Which is the projector to the two dimensional subspace Hu of the Hilbert space.

Page 15

DYNAMICS OF VACUA . ..15
The Weyl symbols of operators with bounded square-trace to which undoubtedly
belong I2and σαcan be found by a direct formula,
?
Technically, one can write the matrices in the (nonorthogonal) basis of |±u/2?
and use the Weyl symbols for the following operators,
|u/2??u/2| ∼ 2e−2|z−u/2|2,
|−u/2??−u/2| ∼ 2e−2|z+u/2|2,
|u/2??−u/2| ∼ 2e−2|z|2+(¯ zu−z¯ u),
|−u/2??u/2| ∼ 2e−2|z|2−(¯ zu−z¯ u),
which can be easily computed.
The sigma-matrices are expressed in the nonorthogonalbasis of |±u/2? as follows,
1
1 − e−|u|2(|u/2??u/2| + |−u/2??−u/2|) +
e
1−e−|u|2(|u/2??−u/2|+|−u/2??u/2|)
σ1=
√1 − e−|u|2(|u/2??u/2| − |−u/2??−u/2|),σ2
X ∼
d¯kdk ei(¯kz+k¯ z)trXe−i(¯ka+k¯ a).
(50)
(51)
σ0=
−1
2|u|2
1
=
1
√1 − e−|u|2(|u/2??−u/2|− |−u/2??u/2|),σ3= −
e−1
1 − e−|u|2|u|2
Inserting (51) into (52) one finds immediately the functions (27) of the third
section.
The plots of functions σα(¯ z,z) can be seen on the fig.3.
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