# nLab vacuum

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## Surveys, textbooks and lecture notes

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• Axiomatizations

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• -theorem

• Tools

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• Structural phenomena

• Types of quantum field thories

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• examples

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# Contents

## Idea

### In quantum field theory

In quantum field theory a vacuum state is supposed to be a quantum state that expresses the absence of any particle excitations of the fields.

On Minkowski spacetime the vacuum state for a free field theory is the standard Hadamard state. On general globally hyperbolic spacetimes there are always Hadamard states, and they do play the role of the vacuum state in the construction of AQFT on curved spacetimes, see at locally covariant perturbative AQFT. Notably the choice of such a Hadamard state fixes the Feynman propagator, hence the time-ordered product of quantum observables and thus the perturbative S-matrix away from coinciding interaction points (the extension of these distributions to coinciding interaction points is the process of renormalization).

However, since on a general globally hyperbolic spacetime there is no globally well-defined concept of particles, there is in general no concept of vacuum state. But under good conditions (such as existence of suitable timelike Killing vectors) one may identify Hadamard states which deserve to be thought of as vacuum states (Brum-Fredenhagen 13).

### In string theory

As mentioned above, in perturbative quantum field theory a choice of vacuum state (more generally: Hadamard state) defines the Feynman propagator, hence the time-ordered product of quantum observables and thus the perturbative S-matrix away from coinciding interaction points and extended to coincing interaction points via a choice of renormalization.

Now perturbative string theory is not a local field theory, in which the S-matrix is derived (up to choice of vacuum state and renormalization) from a local Lagrangian density, but is defined by declaring the string perturbation series (obtained by other means, namely by 2d CFT correlators) to be the scattering matrix.

Hence while a vacuum in perturbative string theory cannot be a quantum state as in local field theory (since there is a priori just no free field Wick algebra that it could be a quantum state on), it makes sense to define a vacuum of perturvative string theory to be whatever data it takes to define the stringy S-matrix, namely the string perturbation series. This datum is that of a 2d SCFT (of central charge 15).

Hence in perturbative string theory the vacua of the effective background QFT are at the same time identified with certain 2-dimensional CFTs – which for “geometric vacua” are the string sigma-models which have the given effective QFT as their second quantization.

Perturbative string theory is defined as the string perturbation series of these sigma-models about these vacua.

The moduli space of these vacua – which is hardly understood – has come to be called the landscape of string theory vacua .

## Examples

Last revised on April 5, 2018 at 04:52:09. See the history of this page for a list of all contributions to it.