# Presentation: 768778

## The fate of the -vacuum

- The fate of the **-vacuum

- Hael Collins
- Carnegie Mellon University
**The de Sitter Days Workshop, Fermilab**
**Saturday, November 15, 2003**

**in collaboration with**
**Rich Holman & Matt Martin**
**hep-th/0306028, hep-ph/0309265, hep-th/0311nnn**

## Overview

**The vacua of de Sitter space**
**The linear divergences of an interacting theory in an **********-vacuum**
**Taming the divergences of the **********-vacuum **
**Truncated **********-vacua and the CMB (next talk)**

## The geometry of de Sitter space

- The cosmological importance of de Sitter space challenges us to understand properly quantum field theory in a de Sitter background
**no global time-like Killing vector**
- Conformally flat coordinates:

**What are the de Sitter**
**invariant vacua?**

## The vacua of de Sitter space

- One vacuum of de Sitter
- 	space has special properties:
- Consider a free scalar field:

**thermal,**
**becomes the flat vacuum at short distances, ...**

*U**k**E*(**)** is defined by**

- The Euclidean vacuum

**Solves the Klein-Gordon Equation**
**Matches flat vacuum at short distances**
**But it is not the unique de Sitter invariant vacuum**

- The **-vacuum

**Mottola, PRD 31, 754 (1985)**
**Allen, PRD 32, 3136 (1985)**

##  -vacuum

## Trouble with the -vacua?

- Trouble with the **-vacua?

- Perturbation theory in
- the **-vacua:

- Einhorn and Larsen found pinched singularities at one-loop order
**hep-th/0209159**
- Banks and Mannelli showed the **-vacua need non-local counterterms
**hep-th/0209113**
- Variants of the **-vacua can be renormalized:
- Place sources at *x* and *x**A*
- Goldstein and Lowe
**hep-th/0302050, hep-th/0308135**
- Einhorn and Larsen
**hep-th/0305056**

## The fate of the -vacuum in an interacting theory

**When we evolve the quantum theory with interactions,**
**we discover a linear divergence in the one-loop corrections**
**which cannot be cancelled by a mass counterterm**
**We write the loop integrals in a form in which the pinch singularities do not appear explicitly**
**cf. pinch singularities in thermal field theory**
**As a check we find that the theory is renormalizable when**
************** ******** – ∞ ****(Euclidean vacuum) with interaction****, ****or**
** ********** ≠ – ∞  ****but with no interactions**

## The Schwinger-Keldysh formalism

- As the background
- 	is time-dependent,
- 	it is dangerous to
- 	ask about the
- 	asymptotic states
- We use a quantization procedure that allows a time-dependent evolution
**Schwinger, J. Math. Phys 2, 407 (1961)**
**Keldych, JETP 20, 1018 (1965)**
- Effectively double the field content of the theory:

**For the Closed Time Contour**
**Double the vertices:**
**use ********+**** and ********–**** vertices**
**Four propagators:**
**G****++****, G****+–****, G****–+****, G****– –**
**Events on the + contour occur before those on the – contour**

## Divergences in the -vacuum

- Divergences in the **-vacuum

**-Wightman function:

- Loop propagator
- Power counting
- Phase interference
- Possible divergences in
- two-propagator loops
- three-propagator loops

## Evolution of the number operator

- Example:
- the number of Euclidean particles in the **-vacuum
- in **-state at **0
- 3 interaction
- Evaluate to one-loop order

## Origin of the divergence

- Define an ‘initial occupation number’
- The **-dependent coefficient of the divergence is thus

**linearly**
**divergent**

**If this were a thermal system these *****n******p*********** s would have a Boltzmann suppression at large *****p***

**However, an **********-state is populated**
**to arbitrarily high momenta**

## -vacuum

## What went wrong with the -vacuum?

- What went wrong with the **-vacuum?

- Since the Euclidean vacuum matches with the flat vacuum, it is reasonable to define the usual time-ordering
- But is it the correct prescription for the **-vacuum?
- antipodinal information
- interference
- Can we generalize the idea of time-ordering
- distinct from Euclidean case
- has a good **  – ∞ limit

**?**

## Propagation in the -vacuum

- Propagation in the **-vacuum

- Define a new time-ordering for the **-vacuum to remove constructive interference in products of propagators
- Up to constants, we obtain an essentially unique form
- contains two point sources

**Goldstein and Lowe, hep-th/0302050, hep-th/0308135**
**Einhorn and Larsen, hep-th/0305056 **

## Note that the time-ordering operator acts on the fields
To disentangle the inherent correlated behavior at the antipodes, T also depends on the antipodes

- To disentangle the inherent correlated behavior at the antipodes, *T*** also depends on the antipodes

- Time ordering in the **-vacuum

- It is useful to have a path integral definition for the field theory

## Since the propagator contains two sources, let us define the generating functional to have two currents:

Fix a and b by differen-tiating with respect to the current
Complete the square in W0

- Fix *a*** and *b*** by differen-tiating with respect to the current
- Complete the square in *W*0**

- The path integral:  free theory

## Express W in terms of W0
Define a new functional derivative
An n-point Green’s function contains three types of propagators
Alpha (both external)
Mixed (one point external)
Euclidean (internal)

- Express *W*** in terms of *W*0**
- Define a new functional derivative
- An *n*-point Green’s function contains three types of propagators
- Alpha (both external)
- Mixed (one point external)
- Euclidean (internal)

- The path integral:  local interactions

## Only Euclidean-vacuum propagators appear in the loop

- Local interactions:  renormalizability

- The loop produces a logarithmic divergence which is cancelled by ***m*2

**+**

**=       finite**

## Local interactions are renormalizable because only GEF appears in loops
Was this necessary?
Recall the original -vacuum
Euclidean
Double-source 
Antipodinal interactions

- Local interactions are renormalizable because only *G**E**F* appears in loops
- Was this necessary?
- Recall the original **-vacuum
- Euclidean
- Double-source **
- Antipodinal interactions

- The path integral:  antipodinal interactions

*E*

**

## Only -vacuum propagators appear in diagrams

- Antipodinal interactions:  renormalizability

- This loop produces a logarithmic divergence too which is cancelled by ***m*2

**+**

**=       finite**

## Although the propagator contains a peculiar piece which depends on the antipode, the theory is
causal
renormalizable in the self-energy
It is non-local, but in a very constrained form
global coordinates
inflationary patch
Do pathologies appear elsewhere?
e.g. vertex corrections?

- Although the propagator contains a peculiar piece which depends on the antipode, the theory is
- causal
- renormalizable in the self-energy
- It is non-local, but in a very constrained form
- global coordinates
- inflationary patch
- Do pathologies appear elsewhere?
- e.g. vertex corrections?

- Further tests

***x***

***x******A***

## Conclusions

**The **********-vacuum**

**The one loop corrections are linearly divergent for the ********-vacuum**
**These cannot be removed by a mass counterterm**
**Euclidean vacuum loops can be renormalized**

**Loop effects receive a ********/*****H***** enhancement**
**the next talk will describe this result**

**Our definition for the **********-propagator is based on flat space intuition**
**Use renormalizability to guide us**
**Self-energy graphs are no longer linearly divergent**
**local and antipodinal interactions**

**Loops and trans-planckian physics**

**An **********-vacuum with two sources**