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Covariant string quantization in Java Get Code39 in Java Covariant string quantization




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Covariant string quantization using none tomake none on asp.net web,windows application .NET Framework In the Lorentz covariant quantization of string theory we treat all string coordinates XJ-l(r, a) on the same footing. To select physical states we use the constraints generated by a subset of the Virasoro operators. The states automatically carry time labels, so the Hamiltonian does not generate time evolution.

We describe the Polyakov string action and show that it is classically equivalent to the Nambu-Goto action.. Introduction In this book, the quantization of strings was carried out using light-cone coordinates and the light-cone gauge. String theory is a Lorentz invariant theory, but Lorentz symmetry is not manifest in the light-cone quantum theory. Indeed, the choice of a particular coordinate X+ for special treatment hides from plain view the Lorentz symmetry of the theory.

While hidden, the Lorentz symmetry is still a symmetry of the quantum theory, as we demonstrated by the construction ofthe Lorentz generator M- 1 This generator has the expected properties when the spacetime has the critical dimension. Since Lorentz symmetry is of central importance, it is natural to ask if we can quantize strings preserving manifest Lorentz invariance. It is indeed possible to do so.

The Lorentz covariant quantization has some advantages over the light-cone quantization. Our lightcone quantization of open strings did not apply to DO-branes because the light-cone gauge requires that at least one spatial open string coordinate have Neumann boundary conditions. Covariant quantization applies to DO-branes.

The equations of motion for the fields that arise in string theory are better understood in Lorentz covariant notation. The calculation of tachyon potentials, alluded to in Section 12.8, appears to be possible only within the Lorentz covariant quantization of strings.

Why then, have we waited so long to discuss the Lorentz covariant quantization of strings The covariant approach is very elegant but it is sometimes hard to extract the physical content from its equations. Moreover, covariant quantization has a series of features that are quite strange. We are accustomed to the idea that in quantum mechanics the position of a particle becomes an operator while time remains a parameter.

In a Lorentz invariant quantization, all the coordinates Xll of a particle, including xo, become operators. Similar remarks apply to the string coordinates XJ-l. We will also see that the string Hamiltonian annihilates the physical states of the theory.

Finally, in covariant quantization it is necessary to discuss states whose norm is not positive; this takes us out of the usual Hilbert space postulates.. Covariant string quantization We chose to do th e quantization of strings in the light-cone gauge because all of the above features would have distracted us from the task of extracting the physical content of the theory. The light-cone gauge has served us well, and it will continue to do so in the following chapters, where we use light-cone string diagrams to begin our discussion of string interactions. The proper treatment of covariant quantization requires tools that go beyond the level of this book.

We will not be able to derive the critical dimension, for example. While our treatment will not be complete, we will still gain some important insights into the structure of the theory. Let us begin our discussion by recalling some facts about parameterizations of the worldsheet.

In 9 we described a large class of gauges characterized by a vector nJ1.. By choosing nJ1.

appropriately we could produce the static or light-cone gauge. For open strings the choice of nJ1. completely fixes the parameterization of the world-sheet.

This is almost true for closed strings as well, except that we are still free to shift the coordinate a rigidly along the strings. We showed that for any choice of nJ1. the string coordinates satisfy the constraints (21.

1) Since any choice of nJ1. results in these constraints, the constraints by themselves do not completely fix the parameterization of the world-sheet. In fact, as you may have seen in Problem 12.

9, many reparameterizations preserve these constraints. We were able to show that the constraints (21.1) cause the equations of motion to become simple wave equations:.

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