This repository was archived by the owner on May 5, 2022. It is now read-only.
-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy paththesis_main.lyx
1857 lines (1398 loc) · 37.1 KB
/
thesis_main.lyx
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
#LyX 2.2 created this file. For more info see http://www.lyx.org/
\lyxformat 508
\begin_document
\begin_header
\save_transient_properties true
\origin unavailable
\textclass jcuthesis
\begin_preamble
%\usepackage[draft,subsubsection]{progress}
\usepackage{braket}
\usepackage[force]{feynmp-auto}
\usepackage{tikz}
\usetikzlibrary{calc,decorations.markings}
\end_preamble
\options sort&compress
\use_default_options true
\begin_modules
theorems-ams
theorems-ams-extended
theorems-sec
\end_modules
\maintain_unincluded_children false
\language australian
\language_package default
\inputencoding utf8
\fontencoding global
\font_roman "default" "default"
\font_sans "default" "default"
\font_typewriter "default" "default"
\font_math "auto" "auto"
\font_default_family default
\use_non_tex_fonts false
\font_sc false
\font_osf false
\font_sf_scale 100 100
\font_tt_scale 100 100
\graphics default
\default_output_format default
\output_sync 0
\bibtex_command default
\index_command default
\paperfontsize 11
\spacing onehalf
\use_hyperref true
\pdf_bookmarks true
\pdf_bookmarksnumbered false
\pdf_bookmarksopen false
\pdf_bookmarksopenlevel 1
\pdf_breaklinks false
\pdf_pdfborder false
\pdf_colorlinks false
\pdf_backref false
\pdf_pdfusetitle true
\papersize default
\use_geometry true
\use_package amsmath 1
\use_package amssymb 1
\use_package cancel 1
\use_package esint 1
\use_package mathdots 1
\use_package mathtools 1
\use_package mhchem 1
\use_package stackrel 1
\use_package stmaryrd 1
\use_package undertilde 1
\cite_engine natbib
\cite_engine_type numerical
\biblio_style plain
\use_bibtopic false
\use_indices false
\paperorientation portrait
\suppress_date false
\justification true
\use_refstyle 1
\index Index
\shortcut idx
\color #008000
\end_index
\leftmargin 35mm
\topmargin 25mm
\rightmargin 25mm
\bottommargin 25mm
\secnumdepth 3
\tocdepth 3
\paragraph_separation indent
\paragraph_indentation default
\quotes_language english
\papercolumns 1
\papersides 1
\paperpagestyle default
\tracking_changes false
\output_changes false
\html_math_output 0
\html_css_as_file 0
\html_be_strict false
\end_header
\begin_body
\begin_layout Title
Symmetry improvement techniques for non-perturbative quantum field theory
\end_layout
\begin_layout Author
Michael Jonathan Brown
\end_layout
\begin_layout Degree
Doctor of Philosophy (Physics)
\end_layout
\begin_layout School
College of Science and Engineering
\end_layout
\begin_layout University
James Cook University
\end_layout
\begin_layout CrestFilename
jculogo.pdf
\end_layout
\begin_layout Acknowledgements
I would like to thank my primary advisor, Prof.
Ian Whittingham, and secondary advisor A/Prof.
Daniel Kosov for their endless patience, encouragement, feedback and support.
\end_layout
\begin_layout Acknowledgements
I would also like to thank my friends and family for their love and support.
Special thanks go to my wife, without whose regular encouragement this
project would have been impossible, and to my parents for encouraging me
in the sciences (though I ended up changing my thesis topic from my father's
suggestion of basket weaving).
\end_layout
\begin_layout Acknowledgements
Thanks to Elwood, who is a glorious bastard.
\end_layout
\begin_layout Acknowledgements
Numerous open source tools were used to build this thesis: \SpecialChar LyX
, \SpecialChar LaTeX2e
, Python, numpy/sci
py/matplotlib, git, and more.
It is good to live in a world with such a vibrant open source community.
Thanks to all who participate.
\end_layout
\begin_layout Acknowledgements
This research was supported by an Australian Postgraduate Award 2013-2016.
\end_layout
\begin_layout Acknowledgements
\begin_inset VSpace vfill
\end_inset
\end_layout
\begin_layout Acknowledgements
\begin_inset Box Frameless
position "t"
hor_pos "c"
has_inner_box 1
inner_pos "t"
use_parbox 0
use_makebox 0
width "100col%"
special "none"
height "1in"
height_special "totalheight"
thickness "0.4pt"
separation "3pt"
shadowsize "4pt"
framecolor "black"
backgroundcolor "none"
status open
\begin_layout Plain Layout
This work is licensed under a
\begin_inset CommandInset href
LatexCommand href
name "Creative Commons Attribution-ShareAlike 4.0 International License"
target "https://creativecommons.org/licenses/by-sa/4.0/"
\end_inset
.
All source code for this thesis document and related programs (
\noun on
Mathematica
\noun default
notebooks etc.) will be made available online at
\begin_inset CommandInset href
LatexCommand href
name "https://github.com/mikejbrown/thesis"
target "https://github.com/mikejbrown/thesis"
\end_inset
.
This document has been built from the following version of the source:
\begin_inset CommandInset include
LatexCommand input
preview true
filename "version"
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Abstract
Quantum field theory is the mathematical language of nature at the deepest
levels currently known.
Viewed in this light, much of the last century of theoretical physics can
be seen as quantum field theory calculations performed in a variety of
approximations.
However, despite having a very successful standard model of physics and
a phenomenally useful perturbation theory for extracting predictions from
it, persistent problems remain.
Many of these are due to the inability to calculate in practice what is
calculable in principle.
The difficulty of these problems comes from the importance of a large number
of degrees of freedom, or a strong coupling between degrees of freedom,
or both.
Only in the case of very simple systems or systems close to thermal equilibrium
can their properties be determined with confidence.
The remaining cases require a resummation, or re-organisation, of perturbation
theory.
However,
\emph on
ad hoc
\emph default
resummations are problematic because perturbation series are asymptotic
in nature and the re-organisation of the terms of an asymptotic series
is mathematically questionable.
Systematic resummation schemes are required to guarantee consistency with
the original non-perturbative theory.
\end_layout
\begin_layout Abstract
\begin_inset Formula $n$
\end_inset
-particle irreducible effective actions (
\begin_inset Formula $n$
\end_inset
PIEAs;
\begin_inset Formula $n=1,2,3,\cdots$
\end_inset
) are a proven method of resummation which can be thought of as generalisations
of mean field theory which are (a) elegant, (b) general, (c) in principle
exact, and (d) have been promoted for their applicability to non-equilibrium
situations.
These properties make the
\begin_inset Formula $n$
\end_inset
PIEAs compelling candidates for filling the gap between theory and experiment
anywhere the problem is the inability to compute the behaviour of strongly
interacting degrees of freedom or degrees of freedom which are out of thermal
equilibrium.
Unfortunately,
\begin_inset Formula $n$
\end_inset
PIEAs are known to violate the symmetries of a field theory when they are
truncated to a form that can be solved in practice.
This can lead to
\emph on
qualitatively
\emph default
wrong physical predictions as an artefact of the method of calculation.
If one takes the
\begin_inset Formula $n$
\end_inset
PIEA predictions at face value one may reject a physically correct model
on the basis of an incorrect prediction.
Therefore it is critical to gain a better understanding of the symmetry
problem in order for
\begin_inset Formula $n$
\end_inset
PIEAs to be useful in the future.
\end_layout
\begin_layout Abstract
This is where this thesis aims to make a contribution.
This thesis examines the theory of global symmetries in the
\begin_inset Formula $n$
\end_inset
PIEA formalism for quantum field theory with the goal of improving the symmetry
properties of solutions obtained from practical approximation schemes.
Following pedagogical reviews of quantum field theory and
\begin_inset Formula $n$
\end_inset
PIEAs, symmetries are introduced by considering a scalar
\begin_inset Formula $\left(\phi^{2}\right)^{2}$
\end_inset
field theory with
\begin_inset Formula $\mathrm{O}\left(N\right)$
\end_inset
symmetry and spontaneous symmetry breaking.
The symmetries are embodied by Ward identities which relate the various
correlation functions of the theory.
When truncated, the
\begin_inset Formula $n$
\end_inset
PIEAs for
\begin_inset Formula $n>1$
\end_inset
violate these identities with phenomenological consequences which are reviewed.
The symmetry improvement method (SI) of
\begin_inset CommandInset citation
LatexCommand citet
key "Pilaftsis2013"
\end_inset
addresses this by imposing the 1PI Ward identities directly onto the solutions
of
\begin_inset Formula $n$
\end_inset
PI equations of motion through the use of Lagrange multipliers.
In novel work, SI is extended to the 3PIEA.
The SI-3PIEA is then renormalised in the Hartree-Fock and two loop truncations
in four dimensions and in the three loop truncation in three dimensions.
The Hartree-Fock equations of motion are solved and compared to the unimproved
and SI-2PI Hartree-Fock solutions.
In both the SI-2PI and SI-3PI methods Goldstone's theorem is satisfied.
However, the SI-2PI solution correctly predicts that the phase transition
is second order while the SI-3PI solution predicts a weak (weaker than
the unimproved 2PIEA) first order transition.
Further checks of the formalism are performed: it is shown that the SI-3PIEA
obeys the Coleman-Mermin-Wagner theorem and is consistent with unitarity
to
\begin_inset Formula $\mathcal{O}\left(\hbar\right)$
\end_inset
, but only if the full three loop truncation is kept.
\end_layout
\begin_layout Abstract
Following this a novel method of soft symmetry improvement (SSI) is introduced.
SSI differs from SI in that the Ward identities are imposed softly in the
sense of least squared error.
A new
\emph on
stiffness
\emph default
parameter controls the strength of the constraint.
The motivation for this method is that the singular nature of the constraint
in SI leads to pathological behaviour such as the non-existence of solutions
in certain truncations and the breakdown of the theory out of equilibrium.
In order to regulate the IR behaviour, a box of finite volume is used and
the limit of infinite volume taken carefully.
Three limiting regimes are found.
Two are equivalent to the unimproved 2PIEA and SI-2PIEA respectively.
The third is a novel limit which has pathological behaviour including a
strongly first order phase transition and a regime where solutions cease
to exist for a range of temperatures below the critical temperature.
As the stiffness increases this range increases and at a finite value of
the stiffness the solution ceases to exist even at zero temperature.
This loss of solution is confirmed both analytically and numerically.
\end_layout
\begin_layout Abstract
Finally, the linear response of a system in equilibrium subject to external
perturbations is studied for the first time in the SI-2PIEA formalism.
It is shown that, beyond equilibrium, the scheme is inconsistent in that
the equations of motion are generically over-constrained.
Thus, the SI-2PIEA predicts qualitatively incorrect physics out of equilibrium.
This result can be understood as a result of the decoupling of the propagator
fluctuation from the field fluctuation in the 2PIEA formalism and the failure
of higher order Ward identities.
This argument generalises so that, as a result, SI-
\begin_inset Formula $n$
\end_inset
PIEA are invalid out of equilibrium, at least within the linear response
approximation in generic truncations.
\end_layout
\begin_layout Standard
\begin_inset CommandInset toc
LatexCommand tableofcontents
\end_inset
\end_layout
\begin_layout Standard
\begin_inset FloatList table
\end_inset
\end_layout
\begin_layout Standard
\begin_inset FloatList figure
\end_inset
\end_layout
\begin_layout Chapter*
List of Publications
\end_layout
\begin_layout Standard
This thesis is largely based on work published by the author (M.J.
Brown) in the following peer reviewed journal articles:
\end_layout
\begin_layout Itemize
M.J.
Brown and I.
B.
Whittingham, Symmetry improvement of 3PI effective actions for O(N) scalar
field theory, Phys.
Rev.
D 91, 085020 (2015).
DOI:
\begin_inset CommandInset href
LatexCommand href
name "10.1103/PhysRevD.91.085020"
target "http://dx.doi.org/10.1103/PhysRevD.91.085020"
\end_inset
, arXiv:
\begin_inset CommandInset href
LatexCommand href
name "1502.03640"
target "http://arxiv.org/abs/1502.03640"
\end_inset
,
\begin_inset CommandInset citation
LatexCommand citep
key "Brown2015a"
\end_inset
,
\end_layout
\begin_deeper
\begin_layout Standard
\emph on
Statement of author contributions:
\emph default
The authors jointly developed the research question.
All initial calculations, figures and the first draft of the manuscript
were produced by Brown.
The manuscript was revised with checking of results and editorial input
from Whittingham.
\end_layout
\end_deeper
\begin_layout Itemize
M.
Brown and I.
Whittingham, Two-particle irreducible effective actions versus resummation:
Analytic properties and self-consistency, Nucl.
Phys.
B900, 477 (2015).
DOI:
\begin_inset CommandInset href
LatexCommand href
name "10.1016/j.nuclphysb.2015.09.021"
target "http://dx.doi.org/10.1016/j.nuclphysb.2015.09.021"
\end_inset
, arXiv:
\begin_inset CommandInset href
LatexCommand href
name "1503.08664"
target "http://arxiv.org/abs/1503.08664"
\end_inset
,
\begin_inset CommandInset citation
LatexCommand citep
key "Brown2015"
\end_inset
,
\end_layout
\begin_deeper
\begin_layout Standard
\emph on
Statement of author contributions:
\emph default
The authors jointly developed the research question.
All initial calculations, figures and the first draft of the manuscript
were produced by Brown.
The manuscript was revised with checking of results and editorial input
from Whittingham.
\end_layout
\end_deeper
\begin_layout Itemize
M.J.
Brown, I.
B.
Whittingham and D.
S.
Kosov, Linear response theory for symmetry improved two particle irreducible
effective actions, Phys.
Rev.
D 93, 105018 (2016).
DOI:
\begin_inset CommandInset href
LatexCommand href
name "10.1103/PhysRevD.93.105018"
target "http://dx.doi.org/10.1103/PhysRevD.93.105018"
\end_inset
, arXiv:
\begin_inset CommandInset href
LatexCommand href
name "1603.03425"
target "http://arxiv.org/abs/1603.03425"
\end_inset
,
\begin_inset CommandInset citation
LatexCommand citep
key "Brown_2016"
\end_inset
,
\end_layout
\begin_deeper
\begin_layout Standard
\emph on
Statement of author contributions:
\emph default
The authors jointly developed the research question.
All initial calculations and the first draft of the manuscript were produced
by Brown.
The manuscript was revised with checking of results and editorial input
from Whittingham and Kosov.
\end_layout
\end_deeper
\begin_layout Itemize
M.J.
Brown and I.
B.
Whittingham, Soft symmetry improvement of two particle irreducible effective
actions, Phys.
Rev.
D 95, 025018 (2017).
DOI:
\begin_inset CommandInset href
LatexCommand href
name "10.1103/PhysRevD.95.025018"
target "http://dx.doi.org/10.1103/PhysRevD.95.025018"
\end_inset
arXiv:
\begin_inset CommandInset href
LatexCommand href
name "1611.05226"
target "https://arxiv.org/abs/1611.05226"
\end_inset
,
\begin_inset CommandInset citation
LatexCommand citep
key "Brown2016"
\end_inset
,
\end_layout
\begin_deeper
\begin_layout Standard
\emph on
Statement of author contributions:
\emph default
The authors jointly developed the research question.
All initial calculations and the first draft of the manuscript were produced
by Brown.
The manuscript was revised with checking of results and editorial input
from Whittingham.
\end_layout
\end_deeper
\begin_layout Standard
\begin_inset FormulaMacro
\newcommand{\lhook}[1]{#1^{\ulcorner}}
\end_inset
\end_layout
\begin_layout Standard
\begin_inset FormulaMacro
\newcommand{\rhook}[1]{#1^{\urcorner}}
\end_inset
\end_layout
\begin_layout Standard
\begin_inset CommandInset include
LatexCommand include
filename "notation.lyx"
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Newpage clearpage
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Float table
wide true
sideways true
status open
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "tab:list-of-calculations"
\end_inset
Summary of
\begin_inset Formula $\mathrm{O}\left(N\right)$
\end_inset
model calculations performed or discussed in the thesis.
\end_layout
\end_inset
\begin_inset Tabular
<lyxtabular version="3" rows="12" columns="7">
<features booktabs="true" tabularvalignment="middle">
<column alignment="center" valignment="top" width="4.5cm">
<column alignment="center" valignment="top" width="4cm">
<column alignment="center" valignment="top" width="3cm">
<column alignment="center" valignment="top" width="2cm">
<column alignment="center" valignment="top" width="2.5cm">
<column alignment="center" valignment="top" width="3.5cm">
<column alignment="center" valignment="top" width="4cm">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Calculation
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Location
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Broken phase Goldstone mass
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Order of phase transition
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Critical temperature
\end_layout
\begin_layout Plain Layout
(leading order)
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Higgs decay rate
\end_layout
\begin_layout Plain Layout
(leading order)
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Comments
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Benchmark
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $0$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $2^{\text{nd}}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\sqrt{\frac{12\bar{v}^{2}}{\hbar\left(N+2\right)}}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\frac{N-1}{2}\frac{\hbar}{16\pi m_{H}}\left(\frac{\lambda v}{3}\right)^{2}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
2PI Hartree-Fock
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Section
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Renormalisation-and-solution-2PI-HF"
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $>0$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $1^{\mathrm{st}}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
correct
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $0$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
2PI External propagator
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Section
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:The-External-Propagator"
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $0$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $1^{\mathrm{st}}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
correct
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
incorrect (violations of unitarity)
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text