Motivation for the study of spacetime noncommutativity comes primarily from its possible use in investigations of (Planck-scale) spacetime fuzziness, but most work focuses on S-matrix/field-theory observables and still very little has been established for geometric observables. We argue that it might be useful to exploit the "pregeometric" formulation of spacetime noncommutativity, which in particular describes the coordinates of the Moyal plane in terms of the phase-space coordinates of a point particle "living" in a auxiliary/fictitious spacetime ($\{x_1,x_2\}_{Moyal} \equiv \{q,p\}_{particle}$). This leads us straightforwardly to a distance operator for the Moyal plane, and allows us to expose some limitations of a previous attempt to describe the "area of a disc" in the Moyal plane. We also observe that from our pregeometric perspective it is rather natural to contemplate a spacetime whose pregeometric picture is based on the phase-space coordinates (fields) of a string. The fact that such "stringspaces" essentially provide spacetime points with extendedness is relevant for the fuzziness of geometric spacetime observables in ways that we preliminarily characterize through an analysis of the distance observable on a two-dimensional stringspace and through the observation that by implementing the Amati-Ciafaloni-Veneziano/Gross-Mende uncertainty relation in the phase space of the pregeometric string one could have stringspaces with a novel type of fuzzy coordinates.
The Moyal star formulation of string field theory is reviewed. The various versions of the star product are compared and related to one another in a regulated theory that resolves associativity anomalies. A summary of computations and challenges is given.
We study Witten open string field theory in the pp-wave background in the tensionless limit, and construct the N-string vertex in the basis which diagonalizes the string perturbative spectrum. We found that the Witten *-product can be viewed as infinite copies of the Moyal product with the same noncommutativity parameter $θ=2$. Moreover, we show that this Moyal structure is universal in the sense that, written in the string bit basis, Witten's *-product for any background can always be given in terms of the above-mentioned Moyal structure. We identify some projective operators in this algebra …
Stimulated by the importance of noncommutative geometry in recent developments in string theory, D-branes and integrable systems, one intends in this work to present a new insight towards adapting the famous idea of Zeeman effect to noncommutativity à la Moyal and develop an analysis leading to connect our results to the Bigatti-Suskind (BS) formulation.
We illustrate a basic framework for analytic computations of Feynman graphs using the Moyal star formulation of string field theory. We present efficient methods of computation based on (a) the monoid algebra in noncommutative space and (b) the conventional Feynman rules in Fourier space. The methods apply equally well to perturbative string states or nonperturbative string states involving D-branes. The ghost sector is formulated using Moyal products with fermionic (b,c) ghosts. We also provide a short account on how the purely cubic theory and/or VSFT proposals may receive some clarification…
In this paper we show that in the presence of an anti-symmetric tensor $B$-background, Witten's star algebra for open string fields persists to possess the structure of a direct product of commuting Moyal pairs. The interplay between the noncommutativity due to three-string overlap and that due to the $B$-background is our main concern. In each pair of noncommutative directions parallel to the $B$-background, the Moyal pairs mix string modes in the two directions and are labeled, in addition to a continuous parameter, by {\it two} discrete values as well. However, the Moyal parameters are $B$-…
We explicitly find the spectrum of the operators $M^{rs}$ and $\widetilde{M}^{rs}$, which specify the star-product in the matter and ghost sectors correspondingly. Further we derive the diagonal representation for the 3-string vertices. Using this representation we identify the appearing Moyal structures in the matter sector. In addition to the continuous non-commutativity parameter $θ(κ)$ found in hep-th/0202087 we find the discrete non-commutativity parametrized by $θ_ξ$.
We consider a diagonalization of Witten's star product for a ghost system of arbitrary background charge and Grassmann parity. To this end we use a bosonized formulation of such systems and a spectral analysis of Neumann matrices. We further identify a continuous Moyal product structure for a combined ghosts+matter system. The normalization of multiplication kernel is discussed.
We establish that the open string star product in the zero momentum sector can be described as a continuous tensor product of mutually commuting two dimensional Moyal star products. Let the continuous variable $κ\in [~0,\infty)$ parametrize the eigenvalues of the Neumann matrices; then the noncommutativity parameter is given by $θ(κ) =2\tanh(πκ/4)$. For each $κ$, the Moyal coordinates are a linear combination of even position modes, and the Fourier transform of a linear combination of odd position modes. The commuting coordinate at $κ=0$ is identified as the momentum carried by half the string…
Using the Moyal star product, we define open bosonic string field theory carefully, with a cutoff, for any number of string oscillators and any oscillator frequencies. Through detailed computations, such as Neumann coefficients for all string vertices, we show that the Moyal star product is all that is needed to give a precise definition of string field theory. The formulation of the theory as well as the computation techniques are considerably simpler in the Moyal formulation. After identifying a monoid algebra as a fundamental mathematical structure in string field theory, we use it as a too…
We complete the construction of the Moyal star formulation of bosonic open string field theory (MSFT) by providing a detailed study of the fermionic ghost sector. In particular, as in the case of the matter sector, (1) we construct a map from Witten's star product to the Moyal product, (2) we propose a regularization scheme which is consistent with the matter sector and (3) as a check of the formalism, we derive the ghost Neumann coefficients algebraically directly from the Moyal product. The latter satisfy the Gross-Jevicki nonlinear relations even in the presence of the regulator, and when t…
It is shown that Witten's star product in string field theory, defined as the overlap of half strings, is equivalent to the Moyal star product involving the relativistic phase space of even string modes. The string field A(x[σ]) can be rewritten as a phase space field of the even modes $x_{2n},x_{0}, p_{2n}$ where $x_{2n}$ are the positions of the even string modes, and $p_{2n}$ are related to the Fourier space of the odd modes $x_{2n+1}$ up to a linear transformation. The $p_{2n}$ play the role of conjugate momenta for the even modes $x_{2n}$ under the string star product. The split string fo…
By diagonalizing the three-string vertex and using a special coordinate representation the matter part of the open superstring star is identified with the continuous Moyal product of functions of anti-commuting variables. We show that in this representation the identity and sliver have simple expressions. The relation with the half-string fermionic variables in continuous basis is given.
In this paper, we recast the fermionic ghost sector of Witten's open bosonic string field theory in the language of noncommutative field theory. In particular, following the methods of hep-th/0202087, we find that in Siegel gauge Witten's star product roughly corresponds to a continuous tensor product of Clifford Algebras, and we formulate important operators of the theory in this language, notably the kinetic operator of vacuum string field theory and the BRST operator describing the vacuum of the unstable D-25 brane. We find that the BRST operator is singular in this formulation. We explore …
In this paper we study the midpoint structure of the algebra of open strings from the standpoint of the operator/Moyal formalism. We construct a split string description for the continuous Moyal product of hep-th/0202087, study the breakdown of associativity in the star algebra, and identify in infinite sequence of new (anti)commutative coordinates for the star product in in the complex plane. We also explain how poles in the open string non(anti)commutativity parameter correspond to certain ``null'' operators which annihilate the vertex, implying that states proportional to such operators ten…
Besides its various applications in string and D-brane physics, the $θ$-deformation of space (-time) coordinates (naively called the noncommutativity of coordinates), based on the $\star$-product, behaves as a more general framework providing more mathematical and physical informations about the associated system. Similarly to the Gelfand-Dickey framework of pseudo differential operators, the Moyal $θ$-deformation applied to physical problems makes the study more systematic. Using these facts as well as the backgrounds of Moyal momentum algebra introduced in previous works [21, 25, 26], we loo…
We compute the Moyal phase factor for open strings ending on D3-branes wrapping a NSNS Melvin universe in a decoupling limit explicitly using world sheet formalism in cylindrical coordinates.
We report on new results in Witten's cubic string field theory for the off-shell factor in the 4-tachyon amplitude that was not fully obtained explicitly before. This is achieved by completing the derivation of the Veneziano formula in the Moyal star formulation of Witten's string field theory (MSFT). We also demonstrate detailed agreement of MSFT with a number of on-shell and off-shell computations in other approaches to Witten's string field theory. We extend the techniques of computation in MSFT, and show that the j=0 representation of SL(2,R) generated by the Virasoro operators $L_{0},L_{\…
The interaction vertex for a fermionic first order system of weights (1,0) such as the twisted bc-system, the fermionic part of N=2 string field theory and the auxiliary ηξsystem of N=1 strings is formulated in the Moyal basis. In this basis, the Neumann matrices are diagonal; as usual, the eigenvectors are labeled by κ\in\R. Oscillators constructed from these eigenvectors make up two Clifford algebras for each nonzero value of κ. Using a generalization of the Moyal-Weyl map to the fermionic case, we classify all projectors of the star-algebra which factorize into projectors for each κ-subspac…
Deformation quantization is applied to quantize gravitational systems coupled with matter. This quantization procedure is performed explicitly for quantum cosmology of these systems in a flat minisuper(phase)space. The procedure is employed in a quantum string minisuperspace corresponding to an axion-dilaton system in an isotropic FRW Universe. The Wheeler-DeWitt-Moyal equation is obtained and its corresponding Wigner function is given analytically in terms of Meijer's functions. Finally, this Wigner functions is used to extract physical information of the system.
Deformation theory refers to an apparatus in many parts of math and physics for going from an infinitesimal (= first order) deformation to a full deformation, either formal or convergent appropriately. If the algebra being deformed is that of observables, the result is deformation quantization, independent of any realization in terms of Hilbert space operators. There are very important but rare cases in which a formula for a full deformation is known. For physics, the most important is the Moyal-Weyl star product formula. In this paper, we concentrate on deformations of Courant algebroid struc…
We study scalar field and string theory on non commutative q-deformed spaces. We define a product of functions on a non commutative algebra of functions resulting from the q-deformation analogue to the Moyal product for canonically non commutative spaces. We then give the general procedure to define scalar field and classical string theories on the mentioned spaces, we argue that the resulting theories will have enlarged sets of both spacetime and internal symmetries which can be used to study gravitational effects due to the q-deformation.
We propose an analytic framework to study the nonperturbative solutions of Witten's open string field theory. The method is based on the Moyal star formulation where the kinetic term can be split into two parts. The first one describes the spectrum of two identical half strings which are independent from each other. The second one, which we call midpoint correction, shifts the half string spectrum to that of the standard open string. We show that the nonlinear equation of motion of string field theory is exactly solvable at zeroth order in the midpoint correction. An infinite number of solutio…
The appearance of space/time non-commutativity in theories of open strings with a constant non-diagonal background metric is considered. We show that, even if the space-time coordinates commute, when there is a metric with a time-space component, no electric field and the boundary condition along the spatial direction is Dirichlet, a Moyal phase still arises in products of vertex operators. The theory is in fact dual to the non-commutatitive open string (NCOS) theory. The correct definition of the vertex operators for this theory is provided. We study the system also in the presence of a $B$ f…
Non commutative quantum field theory is a possible candidate for the quantization of gravity. In our thesis we study in detail the $ϕ4$ model on the Moyal plane with an harmonic potential. Introduced by Grosse and Wulkenhaar, this model exhibits the Langmann-Szabo duality not only for the vertex but also for the propagator. We have obtained several results concerning this model. We have proved the renormalisability of this theory at all orders in the position space. We have introduced the parametric and Complete Mellin representation for the model. Furthermore we have proved that the coupling …
Solitonic objects play a central role in gauge and string theory (as, e.g., monopoles, black holes, D-branes, etc.). Certain string backgrounds produce a noncommutative deformation of the low-energy effective field theory, which allows for new types of solitonic solutions. I present the construction, moduli spaces and dynamics of Moyal-deformed solitons, exemplified in the 2+1 dimensional Yang-Mills-Higgs theory and its Bogomolny system, which is gauge-fixed to an integrable chiral sigma model (the Ward model). Noncommutative solitons for various 1+1 dimensional integrable systems (such as sin…
Misner space, also known as the Lorentzian orbifold $R^{1,1}/boost$, is the simplest tree-level solution of string theory with a cosmological singularity. We compute tree-level scattering amplitudes involving twisted states, using operator and current algebra techniques. We find that, due to zero-point quantum fluctuations of the excited modes, twisted strings with a large winding number $w$ are fuzzy on a scale $\sqrt{\log w}$, which can be much larger than the string scale. Wave functions are smeared by an operator $\exp(Δ(ν) \partial_+ \partial_-)$ reminiscent of the Moyal-product of non-co…
A non-associative Groenewold-Moyal plane is constructed using quaternion-valued function algebras. The symmetrized multi-particle states, the scalar product, the annihilation/creation algebra and d the formulation in terms of a Hopf algebra are also developed. Non-associative quantum algebras in terms of position and momentum operators are given as the simplest examples of a framework whose applications may involve string theory and non-linear quantum field theory
We elucidate relations between different approaches to describing the nonassociative deformations of geometry that arise in non-geometric string theory. We demonstrate how to derive configuration space triproducts exactly from nonassociative phase space star products and extend the relationship in various directions. By foliating phase space with leaves of constant momentum we obtain families of Moyal-Weyl type deformations of triproducts, and we generalize them to new triproducts of differential forms and of tensor fields. We prove that nonassociativity disappears on-shell in all instances. W…
On the basis of the intimate relation between Nambu dynamics and the hydrodynamics, the hydrodynamics on a non-commutative space (obtained by the quantization of space), proposed by Nambu in his last work, is formulated as ``hydrodynamics of granular material''. In Part 1, the quantization of space is done by Moyal product, and the hydrodynamic simulation is performed for the so obtained two dimensional fluid, which flows inside a canal with an obstacle. The obtained results differ between two cases in which the size of a fluid particle is zero and finite. The difference seems to come from the…
Four years ago the Extended Scale Relativity (ESR) theory in C-spaces (Clifford manifolds) was proposed as the plausible physical foundations of string theory. In such theory the speed of light and the minimum Planck scale are the two universal invariants. All the dimensions of a C-space can be treated on equal footing by implementing the holographic principle associated with a nested family of p-loops of various dimensionalities. This is achieved by using polyvector valued coordinates in C-spaces that encode in one stroke points, lines, areas, volumes,.....We review the derivation of the mini…
