scholarly journals LOOP VARIABLES AND THE INTERACTING OPEN STRING IN A CURVED BACKGROUND

2005 ◽  
Vol 20 (14) ◽  
pp. 1037-1045
Author(s):  
B. SATHIAPALAN
Keyword(s):  
Gauge Invariance ◽  
Equations Of Motion ◽  
Interaction Strength ◽  
Flat Space ◽  
Open String ◽  
Target Space ◽  
Gauge Invariant ◽  
Free Case ◽  
New Feature ◽  
Generally Covariant

Applying the loop variable proposal to a sigma model (with boundary) in a curved target space, we give a systematic method for writing the gauge and generally covariant interacting equations of motion for the modes of the open string in a curved background. As in the free case described in an earlier paper, the equations are obtained by covariantizing the flat space (gauge invariant) interacting equations and then demanding gauge invariance in the curved background. The resulting equation has the form of a sum of terms that would individually be gauge invariant in flat space or at zero interaction strength, but mix amongst themselves in curved space when interactions are turned on. The new feature is that the loop variables are deformed so that there is a mixing of modes. Unlike the free case, the equations are coupled, and all the modes of the open string are required for gauge invariance.

scholarly journals LOOP VARIABLES AND THE (FREE) OPEN STRING IN A CURVED BACKGROUND

2005 ◽  
Vol 20 (04) ◽  
pp. 227-242 ◽  
Author(s):  
B. SATHIAPALAN
Keyword(s):  
Gauge Invariance ◽  
Sigma Model ◽  
Equations Of Motion ◽  
Flat Space ◽  
Open String ◽  
Target Space ◽  
Systematic Method ◽  
Curved Space ◽  
Generally Covariant ◽  
Space Equation

Using the loop variable formalism as applied to a sigma model in curved target space, we give a systematic method for writing down gauge and generally covariant equations of motion for the modes of the free open string in curved space. The equations are obtained by covariantizing the flat space equation and then demanding gauge invariance, which introduces additional curvature couplings. As an illustration of the procedure, the spin-two case is worked out explicitly.

scholarly journals LOOP VARIABLES AND GAUGE INVARIANCE IN (OPEN) BOSONIC STRING THEORY

2002 ◽  
Vol 17 (18) ◽  
pp. 1175-1190 ◽  
Author(s):  
B. SATHIAPALAN
Keyword(s):  
String Theory ◽  
Gauge Invariance ◽  
Equations Of Motion ◽  
Standard Form ◽  
Open String ◽  
Finite Width ◽  
Massive Mode ◽  
Gauge Transformations ◽  
Variable Approach ◽  
Free Case

We give a simplified and more complete description of the loop variable approach for writing down gauge-invariant equations of motion for the fields of the open string. A simple proof of gauge invariance to all orders is given. In terms of loop variables, the interacting equations look exactly like the free equations, but with a loop variable depending on an extra parameter, thus making it a band of finite width. The arguments for gauge invariance work exactly as in the free case. We show that these equations are Wilsonian RG equations with a finite worldsheet cutoff and that in the ir limit, equivalence with the Callan–Symanzik β-functions should ensure that they reproduce the on-shell scattering amplitudes in string theory. It is applied to the tachyon–photon system and the general arguments for gauge invariance can be easily checked to the order calculated. One can see that when there is a finite worldsheet cutoff in place, even the U(1) invariance of the equations for the photon, involves massive mode contributions. A field redefinition involving the tachyon is required to get the gauge transformations of the photon into the standard form.

Feldtheoretische Konstruktion der Jordan—Brans—Dicke-Theorie / Fieldtheoretic Construction of the Jordan-Brans-Dicke-Theory

1971 ◽  
Vol 26 (4) ◽  
pp. 599-622
Author(s):  
H. von Grünberg
Keyword(s):  
Gauge Group ◽  
Tensor Field ◽  
Mathematical Proof ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Flat Space ◽  
Momentum Tensor ◽  
Field Equations ◽  
Tensor Theory ◽  
Generally Covariant

Abstract In the framework of Lorentz invariant theories of gravitation the fieldtheoretic approach of the generally covariant Jordan-Brans-Dicke-theory is investigated.It is shown that a slight restriction of the gauge group of Einstein's linear tensor theory leads to the linearized Jordan-Brans-Dicke-theory. The problem of the inconsistency of the field equations and the equations of motion is solved by introducing the Landau-Lifschitz energy momentum tensor of the gravitational field as an additional source term into the field equations. The second order of the theory together with the corresponding gauge group are calculated explicitly. By means of the structure of the gauge group of the tensor field it is possible to identify the successive orders of the scalar-tensor theory as an expansion of the Jordan-Brans-Dicke-theory in flat space-time. The question of the uniqueness of the procedure is answered by showing that the structure of the gauge group of the tensor field is predetermined by the linear equations of motion. The mathematical proof of this fact confirms formally the meaning of the equations of motion for the geometry of space.

scholarly journals Exact renormalization group and loop variables: A background independent approach to string theory

2015 ◽  
Vol 30 (32) ◽  
pp. 1530055 ◽  
Author(s):  
B. Sathiapalan
Keyword(s):  
Renormalization Group ◽  
String Theory ◽  
Equations Of Motion ◽  
Flat Space ◽  
World Sheet ◽  
Gauge Invariant ◽  
Variable Approach ◽  
String Loop ◽  
Higher Spin ◽  
Exact Renormalization Group

This paper is a self-contained review of the loop variable approach to string theory. The Exact Renormalization Group is applied to a world sheet theory describing string propagation in a general background involving both massless and massive modes. This gives interacting equations of motion for the modes of the string. Loop variable techniques are used to obtain gauge invariant equations. Since this method is not tied to flat space–time or any particular background metric, it is manifestly background independent. The technique can be applied to both open and closed strings. Thus gauge invariant and generally covariant interacting equations of motion can be written for massive higher spin fields in arbitrary backgrounds. Some explicit examples are given.

scholarly journals ACTION FOR (FREE) OPEN STRING MODES IN AdS SPACE USING THE LOOP VARIABLE APPROACH

2007 ◽  
Vol 22 (02) ◽  
pp. 107-117 ◽  
Author(s):  
B. SATHIAPALAN
Keyword(s):  
Equations Of Motion ◽  
Flat Space ◽  
Open String ◽  
Closed String ◽  
Group Method ◽  
Open Strings ◽  
Variable Approach ◽  
Massive Spin ◽  
Complete Treatment ◽  
Space Case

The loop variable technique (for open strings in flat space) is a gauge-invariant generalization of the renormalization group method for obtaining equations of motion. Unlike the beta functions, which are only proportional to the equations of motion, here it gives the full equation of motion. In an earlier paper, a technique was described for adapting this method to open strings in gravitational backgrounds. However, unlike the flat space case, these equations cannot be derived from an action and are therefore not complete. This is because there are ambiguities in the method that involve curvature couplings that cannot be fixed by appealing to gauge invariance alone but need a more complete treatment of the closed string background. An indirect method to resolve these ambiguities is to require symmetricity of the second derivatives of the action. In general this will involve modifying the equations by terms with arbitrarily high powers of curvature tensors. This is illustrated for the massive spin-two field. It is shown that in the special case of an AdS or dS background, the exact action can easily be determined in this way.

scholarly journals LOOP VARIABLES AND GAUGE INVARIANCE IN CLOSED BOSONIC STRING THEORY

2004 ◽  
Vol 19 (38) ◽  
pp. 2857-2870 ◽  
Author(s):  
B. SATHIAPALAN
Keyword(s):  
Gauge Invariance ◽  
Open String ◽  
Closed String ◽  
Bosonic String ◽  
Tree Level ◽  
Gauge Invariant ◽  
Open Strings ◽  
Invariant Description ◽  
String Amplitudes ◽  
Closed Bosonic String

We extend an earlier proposal for a gauge-invariant description of off-shell open strings (at tree level), using loop variables, to off-shell closed strings (at tree level). The basic idea is to describe the closed string amplitudes as a product of two open string amplitudes (using the technique of Kawai, Lewellen and Tye). The loop variable techniques that were used earlier for open strings can be applied here mutatis mutandis. It is a proposal for a theory whose on-shell amplitudes coincide with those of the closed bosonic string in 26 dimensions. It is also gauge-invariant off-shell. As was the case with the open string, the interacting closed string looks like a free closed string thickened to a band.

scholarly journals LOOP VARIABLES AND GAUGE INVARIANT INTERACTIONS OF MASSIVE MODES IN STRING THEORY

1996 ◽  
Vol 11 (07) ◽  
pp. 571-585 ◽  
Author(s):  
B. SATHIAPALAN
Keyword(s):  
String Theory ◽  
Equations Of Motion ◽  
Finite Thickness ◽  
Open String ◽  
Operator Product ◽  
Lower Mass ◽  
Gauge Invariant ◽  
Interaction Terms ◽  
Variable Approach

The loop variable approach used earlier to obtain free equations of motion for the massive modes of the open string, is generalized to include interaction terms. These terms, which are polynomial, involve only modes of strictly lower mass. Considerations based on operator product expansions suggest that these equations are particular truncations of the full string equations. The method involves broadening the loop to a band of finite thickness that describes all the different interacting strings. Interestingly. in terms of these variables. the theory appears non-interacting.

FLRW COSMOLOGY FROM YANG–MILLS GRAVITY WITH TRANSLATIONAL GAUGE SYMMETRY

2013 ◽  
Vol 28 (07) ◽  
pp. 1350018 ◽  
Author(s):  
DANIEL KATZ
Keyword(s):  
Gauge Invariance ◽  
Equations Of Motion ◽  
Flat Space ◽  
Metric Tensor ◽  
Yang Mills ◽  
Effective Metric ◽  
Starting Point ◽  
Special Cases ◽  
The Subject ◽  
Flrw Cosmology

We extend to basic cosmology the subject of Yang–Mills gravity — a theory of gravity based on local translational gauge invariance in flat space–time. It has been shown that this particular gauge invariance leads to tensor factors in the macroscopic limit of the equations of motion of particles which plays the same role as the metric tensor of general relativity (GR). The assumption that this "effective metric" tensor takes on the standard FLRW form is our starting point. Equations analogous to the Friedmann equations are derived and then solved in closed form for the three special cases of a universe dominated by (1) matter, (2) radiation and (3) dark energy. We find that the solutions for the scale factor are similar to, but distinct from, those found in the corresponding GR based treatment.

scholarly journals LOOP VARIABLES AND GAUGE-INVARIANT INTERACTIONS — I

2000 ◽  
Vol 15 (30) ◽  
pp. 4761-4795 ◽  
Author(s):  
B. SATHIAPALAN
Keyword(s):  
Infinite Number ◽  
Dimensional Reduction ◽  
Open String ◽  
Space Time ◽  
Finite Width ◽  
Higher Dimension ◽  
Gauge Invariant ◽  
Variable Approach ◽  
Free Case ◽  
Definition Of

We describe a method of writing down interacting equations for all the modes of the bosonic open string. It is a generalization of the loop variable approach that was used earlier for the free, and lowest order interacting cases. The generalization involves, as before, the introduction of a parameter to label the different strings involved in an interaction. The interacting string has thus becomes a "band" of finite width. The interaction equations expressed in terms of loop variables, has a simple invariance that is exact even off shell. A consistent definition of space–time fields requires the fields to be functions of all the infinite number of gauge coordinates (in addition to space–time coordinates). The theory is formulated in one higher dimension, where the modes appear massless. The dimensional reduction that is needed to make contact with string theory (which has been discussed earlier for the free case) is not discussed here.

scholarly journals Background independence, gauge invariance and equations of motion for closed string modes

2015 ◽  
Vol 30 (18n19) ◽  
pp. 1550105 ◽  
Author(s):  
B. Sathiapalan
Keyword(s):  
Renormalization Group ◽  
Equations Of Motion ◽  
Closed String ◽  
World Sheet ◽  
Gauge Invariant ◽  
Background Independence ◽  
The World ◽  
Exact Renormalization Group ◽  
Generally Covariant ◽  
Derivatives Of

In an earlier paper, gauge invariant and background covariant equations for closed string modes were obtained from the exact Renormalization Group of the world sheet theory. The background metric (but not the physical metric) had to be flat and hence the method was not manifestly background independent. In this paper, the restrictions on the background metric are relaxed. A simple prescription for the map from loop variables to space–time fields is given whereby for arbitrary backgrounds the equations are generally covariant and gauge invariant. Extra terms involving couplings of the curvature tensor to (derivatives of) the Stueckelberg fields have to be added. The background metric can thus be chosen to be the physical metric without any restrictions. This method thus gives manifestly background independent equations of motion for both open and closed string modes.

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