We propose a phase-space path integral formulation of noncommutative quantum mechanics, and prove its equivalence to the operatorial formulation. As an illustration, the partition function of a noncommutative two-dimensional harmonic oscillator is calculated.Noncommutative quantum mechanics represents a natural extension of usual quantum mechanics, in which one allows nonvanishing commutators also between the coordinates, and between the momenta. Denoting the coordinates and the momenta collectively by {x i }, in (2+1)-dimensions one has the commutation relations

We propose an ansatz for the equations of motion of the noncommutative model of a tachyonic scalar field interacting with a gauge field, which allows one to find time-dependent solutions describing decaying solitons. These correspond to the collapse of lower dimensional branes obtained through tachyon condensation of unstable brane systems in string theory.

We propose a simple low-energy classical experiment in which the effects of noncommutativity can be clearly separated from commutative physics. The ensuing bound on the noncommutative scale is remarkable, especially in view of its elementary derivation.

We consider a simple modification of standard phase-space path integrals and show that it leads in configuration space to Lagrangians depending also on accelerations.

We discuss examples of systems which can be quantized consistently, although they do not admit a Lagrangian description.Whether a given set of equations of motion admits or not a Lagrangian formulation has been an interesting issue for a long time. As early as 1887, Helmholtz formulated necessary and sufficient conditions for this to happen, and the problem has a rich history [1]. More recently, motivated by some unpublished work of Feynman [2], a connection was made between the existence of a Lagrangian and the commutation relations satisfied by a given system [3,4]. Ref. [3] concluded that under quite general conditions, including commutativity of the coordinates, [q i , q j ] = 0, the equations of motion of a point particle admit a Lagrangian formulation. The purpose of this note is to demonstrate the reverse, namely that noncommutativity of the coordinates forbids a Lagrangian formulation (therefore a Lagrangian implies commutativity). This happens in all but a few cases, which we all identify. On the other hand, an extended Hamiltonian formulation always remains available. It permits quantization of the system in any of the three usual formalisms: operatorial, wave-function, or path integral. Several examples will be used to illustrate the properties of such unusual systems.We work in a (2+1)-dimensional space, although our considerations easily extend to higher dimensions, and assume that [q 1 , q 2 ] = iθ = 0.(1)