Talk at Clare Hall Quantum Cosmology Very early Universe boundary conditions, DeWitt, hartle-hawking, no-boundary, quantum cosmology, vilenkin, wave function of the universe

1. The value of a boundary
Paulo Moniz*
CMA – Physics Department
Universidade da Beira Interior
* sabbatical visitor
@ DAMTP, University of Cambridge
https://www.linkedin.com/in/pvmoniz/

2. (Modern) Quantum Cosmology, late 1960’s
Bryce DeWitt
(Ph.D. student with J. Schwinger)
Quantum gravity and (specific) effects
Prior results (Rosenfeld): a bit fuzzy and not conclusive.
3 seminal papers
(in spite of unkind remarks to the young grad student by W. Pauli, who had failed at it...)
DeWitt seemed a bit religious (from reading his biography); curious ...
Cf. H Weyl’s: “In these days the angel of topology and the devil of abstract algebra fight
for the soul of every individual discipline of mathematics”.
More to follow, keep that quote present.

3. Wheeler-DeWitt equation/constraint (to compute the wave function of the universe, Ψ)
(curiously, DeWitt called it ‘Einstein-Schrodinger’ equation ... ).
Due to its nature, you need boundary conditions
DeWitt conjecture: boundary conditions for the initial state of the
Universe should NOT be imposed ad hoc from the ‘outside’**;
should be (!) consistently extracted just from from the intrinsic algebra.
Elegance, faith (?): “the constraints are everything and everything should be extracted
from that” (quote in paper).
Nature is the only necessary ingredient if one knows how to read from it.
‘search and you shall find’- https://biblehub.com/matthew/7-7.htm
(cf. H Weyl’s quote, so, who’s the devil, where is the angel).
But the equations proved insurmountable; only special simple cases could
be dealt
** Nb. ‘Tunneling’ and
‘no-boundary’ are ‘human-made’
i.e. ad hoc from the ‘outside’ of the theory

4. 70s (and 80s)
particle physics, standard model,
quarks, Higgs, Ws and Zs
General Relativity (mid 70’s…)
analysis, differential geometry, topology
Success of black hole thermodynamics and quantum Hawking radiation.
‘New’ tool: Euclidean (Feynman!) path integral
black holes and quantum effects towards a more fundamental description;
not just QFTCS
So, early 80’s, from black hole singularity to the ‘big bang’
Use quantum mechanics ‘at’ it.

5. ‘Tunnelling’ (A. Vilenkin)
minimalist, ‘linear’
vacua states ‘nucleating’ from a potential barrier or ‘classically’ forbidden zone.
realistic solutions,
likelihood of cosmological inflation without ‘much or no additional’ elements.
However, it was limited to restrictive models and not easy to generalize to other types
of cosmologies.

6. ‘No-boundary’ (Hawking and Hartle)
(Feynamn’s resurging ‘shadow
over’ Schwinger’s...)
...Euclidean path integral method.
Very elegant, aesthetical, based on topology
and geometry (only compact 4-dimensional
geometries are to be used)
Einstein (flashback here!!) 1920-30’s ...the universe should be closed and finite i.e.
sort of ‘compact’ (for other classical - initial conditions
– reasons (faith?)...).
But ... divergences, non-uniqueness, a vast landscape
where to plough from, still fitting with cosmological
inflation, albeit ‘additional elements’ were required.
On the whole, an enormous achievement of
human knowledge, not conclusive and
‘getting’ THE quantum state of the universe.

7. So, two (rival!) boundary conditions (to compute the wave
function of the universe) were advanced.
So; this (i.e., ‘no-boundary’ dominance) goes on and on, from the mid- 80’s until late-
010’s ie 30 years or so…, a few attempts of variance here and there, a few alerts here
and there, …
Nb. And I am not mentioning here other approaches
inc. Loop or even Supersymmetric (stringy…)
quantum cosmology, or others in quantum gravity, more detached in procedure but
worthy too.

8. Recently, another tribe:
in the past 12 to 18 months,
papers v1, v2, v3 @ ArxiV,
in succession. Fascinating.. .
JL Lehners and ‘team’ (Berlin): correct way to compute path integrals (in quantum
cosmology) is not the Euclidean but the … Lorentzian; All was convergent (!), thanks
to the Piccard-Lefschetz framework,
E Witten’s Morse theory,
algebraic topology..
Nb. An intriguing question, still without answer is…
how could the ‘Tunneling’or ‘no-boundary’camps missed
on the Piccard-Lefshetz? Maybe they just went on to plough
(‘longitudinally’) the proposals, not questioning them
(somewhat ‘orthogonally’, sometimes).
And possibly ignored those alerts... . They were there.
So, how about those boundary conditions rivalry?
Just ditch them? Well, not quite.
Twist time!!

9. Arxiv 1903.06757v1: adding (ad hoc!) ingredients e.g. a ‘combination’ of Neumann
and Dirichlet conditions bring the Piccard-Lefschetz and the ‘no-boundary’ working...
But, even better: arXiv:1507.04212, 1412.6439, 1403.2424 demonstrate that, from the
intrinsic algebra (!) [of constraints; cf. DeWitt’s 1967 guideline (flashback here!)],
different boundary conditions can be select, that are NOT imposed ad hoc from the
‘outside’, but they already were ‘there if one looked for them…’
Very simple models
For different matter contents
Boundary conditions can be established from ‘within’ (and NOT ad hoc).
Bring the ‘no-boundary’ within the
Piccard-Lefschetz
(cf. DeWitt’s programme)

10. Many angles, onto a full circle?
Still far, but assembling path integral,
topology, DeWitt’s algebra,
Schwinger-Feynman shadows,
(new) ‘tribes’,
Einstein always ‘there’… .
Although still elusive, if supersymmetry is found, all this would acquire a fascinating
scenario: from Witten’s Piccard-Lefschetz description, one borders … supersymmetry.
All this is truly impassioned and the future can be really,
vehemently intense for mathematics and theoretical physics.
On the whole, it is a mesmerising story, still ongoing, covering
several decades. It crosses ‘agnostic vs ‘non-agnostic, ‘angels’
and ‘demons’ (or ‘devils’...), hard mathematics, physics ‘holy grail’,… from the XX to
XXI century, creative advances from ‘top brass’ rivalry, more in later recent years .
Recall H Weyl’s, ‘In these days the angel of topology
and the devil of abstract algebra fight for the soul
of every individual discipline of mathematics’.
So who has it? Topology? Algebra? Is it to be ‘seek and
shall be found’ or ad hoc? Who’s being an ‘angel’ or who has tracked a ‘devil’?