## The Bose Einstein Condensation Program |

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The following papers are all associated with a program to construct, mathematically rigorously, a model of a Bose gas exhibiting Bose Einstein condensation and symmetry breaking. The collaborators in that program are T. Balaban (Rutgers), J. Feldman (UBC), H. Knörrer (ETH, Zürich) and E. Trubowitz (ETH, Zürich) and all of the papers have that authorship. We have divided the program into a number of stages.

- The Functional Integral Representation
In the first stage, the partition function and correlation
functions of the model are represented as limits of finite
dimensional integrals (with the dimension tending to infinity
in the limit). In each integral there is one (complex)
integration variable for each point in a discrete torus,
that is to be thought of as an approximation to space-time.
This first stage has been completed.
- A Simple High Temperature Expansion
These papers do not constitute one of the stages. Instead, they
concern a high temperature expansion that is heavily used in the
remaining stages.
- The Temporal Ultraviolet Limit
In the second stage, the lattice spacing in the temporal
direction is sent to zero, by using decimation style renormalization
group methods. This second stage has been completed (for the partition
function - correlation functions would be treated similarly).
- The Small Field Part of the Parabolic Flow
In the third stage, a block spin renormalization group flow with
parabolic scaling rules is followed until the running chemical
potential becomes (almost) one. Until that point the fact that the
minimum of the effective potential is not at the origin is not
very important. Under parabolic scaling rules, the scaling in the time
direction is different than the scaling in space directions.
In this third stage, only the small field part of the model
(i.e. the part of the domain of integration near the critical point of
the action) is treated. This third stage has been completed.
- The Small Field Part of the Elliptic Flow In the fourth stage, a block spin renormalization group flow with elliptic scaling rules is followed (after having executed a relatively small number of additional parabolic steps, in order to build the running chemical potential up to a number somewhat larger than one). In this regime the fact that minimum of the effective potential is not at the origin is very important. Under elliptic scaling rules, the same scaling is used in temporal and spatial directions. In this fourth stage, only the small field part of the model (i.e. the part of the domain of integration near the critical point of the action) is treated. This fourth stage is currently under construction.

Annales Henri Poincaré,

[ pdf (275KB), published version (subscription required)]

** A Functional Integral Representation for Many Boson Systems
II: Correlation Functions**.

Annales Henri Poincaré, ** 9**, 1275-1307 (2008).

[ pdf (221KB),
published version (subscription required)]

Journal of Statistical Physics,

pdf (180KB)

** Power Series Representations for Complex Bosonic Effective Actions
I: A Small Field Renormalization Group Step**.

Journal of Mathematical Physics, **51**, 053305 (2010).

[ pdf (309KB),
AIP's official online abstract (subscription required)]

Copyright (2010) American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics.

** Power Series Representations for Complex Bosonic Effective Actions.
II: A Small Field Renormalization Group Flow**.

Journal of Mathematical Physics, **51**, 053306 (2010).

[ pdf (260KB),
AIP's official online abstract (subscription required)]

Copyright (2010) American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics.

** Power Series Representations for Complex Bosonic Effective Actions.
III. Substitution and Fixed Point Equations**.

Annales de l'Institut Henri Poincaré D, ** 6**, 43-71 (2019).

[ preprint (222KB),
published version]

Annales Henri Poincaré,

[ pdf (1430KB), published version (subscription required)]

**The Temporal Ultraviolet Limit**,
Les Houches 2010.

This is a draft of Feldman's Les Houches 2010 course. It provides
more pedagogical treatments of various aspects of the construction.

pdf (605KB)

Annales Henri Poincaré,

[ preprint (307KB), published version]

** The Small Field Parabolic Flow for Bosonic Many-body Models:
Part 1 - Main Results and Algebra**.

Annales Henri Poincaré, ** 20**, 1-62 (2019).

[ preprint (506KB),
published version (view only)]

** The Small Field Parabolic Flow for Bosonic Many--body Models:
Part 2 - Fluctuation Integral and Renormalization**.

Annales Henri Poincaré, ** 20**, 63-124 (2019).

[ preprint (450KB),
published version (view only)]

** The Small Field Parabolic Flow for Bosonic Many--body Models:
Part 3 - Nonperturbatively Small Errors**.

[ preprint (175KB)]

** The Small Field Parabolic Flow for Bosonic Many--body Models:
Part 4 - Background and Critical Field Estimates**.

[ preprint (301KB)]

** Operators for Parabolic Block Spin Transformations**.

[ preprint (448KB)]

** Bloch Theory for Periodic Block Spin Transformations**.

[ preprint (207KB)]

** The Algebra of Block Spin Renormalization Group Transformations**.

[ preprint (176KB)]

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