Barker, William H.

William H. Barker

Bowdoin College
Brunswick, Maine

Books:  Continuous Symmetry: From Euclid to Klein (AMA, 2007)

________ Harmonic Analysis on Reductive Groups

(NOTE: A conference on Harmonic Analysis on Reductive Groups was held at Bowdoin College in Brunswick, Maine from July 31 to August 11, 1989. The stated goal of the conference was to explore recent advances in harmonic analysis on both real and p-adic groups. It was the first conference since the AMS Summer Sym­posium on Harmonic Analysis on Homogeneous Spaces, held at Williamstown, Massachusetts in 1972, to cover local harmonic analysis on reductive groups in such detail and to such an extent. While the Williamstown conference was longer (three weeks) and somewhat broader (nilpotent groups, solvable groups, as well as semisimple and reductive groups), the structure and timeliness of the two meetings was remarkably similar. The program of the Bowdoin Conference consisted of two parts. First, there were six major lecture series, each consisting of several talks addressing those topics in harmonic analysis on real and p-adic groups which were the focus of intensive research during the previous decade. These lectures began at an introductory level and advanced to the current state of research. Sec­ond, there was a series of single lectures in which the speakers presented an overview of their latest research.

_____ Lp harmonic analysis on SL(2,R)


Most recent email: Friday, 7 February 2020

Dear Prof. Dr. William H. Barker:

My work in 1972 focused on continuity, symmetry, and harmony. I was attempting to define what I thought would entail “a moment of perfection” within our quantum universe. By 1980, after working with an array of distinguished scholars in Boston, Cambridge (USA), and Paris, I went back to work within a business that I had started in 1971. From a little service bureau, we soon had a software business with well over 100 employees. My first opportunity to attempt to dig back into it all back was in 2011. I was helping a nephew with his high school geometry classes when we went inside the tetrahedron — — and then its octahedron, step-by-step, deeper and deeper by dividing all the edges by 2 and connecting those new vertices. Within 45 steps we were within particle physics. In 67 additional steps, we were within the Planck scale. By multiplying those classroom objects by 2, in 90 steps we were out to the approximate age and size of the universe. Instead of base-10 like Kees Boeke (1957), we used base-2, we had an inherent geometry, and we went from the Planck units to the current time.

It was an unusual, albeit, rather idiosyncratic chart of 202 notations:

Prima facie, do you see any merit to such a chart?

I will continue my readings of your work, Continuous Symmetry: From Euclid to Klein (AMA, 2007) and Harmonic Analysis on Reductive Groups (Springer, 1991) in hopes that you might have some guiding thoughts for  this rather idiosyncratic chart of the universe.  Thank you.

Most sincerely,


PS. In 1746 our family settled in Bremen, Maine. Bowdoin had always been on my list of schools to consider, but in 1965 the call for voter registration in the South won the day.  I always think of you all on my way out of Freeport and as we go through Brunswick. -BEC

First email: August 2, 2016 3:20 PM

Dear Prof. Dr. William H. Barker:

My grandmother lived up the road a ways (Bremen…Damariscotta, then out to 1A and the coast). Often Dad would stop at Valerie’s in Ogunquit, my sister’s favorite restaurant; they shared the name. We’d fall quickly back to sleep as children for the final long slog up from Cambridge. For some magic reason, I would awake just as we were passing by Bowdoin. Bathed in the soft summer lights, I would secretly dream, “That’ll be my school.”

1965 came quickly and I marched off to the south to register voters, but Bowdoin always held that special place.

Today, I am delighted to find your book on continuous symmetries and remember my childhood once more. Images imprint the soul and make us who we are.

When and why is there spontaneous symmetry breaking?
Have you given it much thought?

So, I have discovered your work and I am grateful to now be taking a de facto course with you through your writing. And so I say, “Thank you!”

With warm regards,
Most sincerely,



How does one find your work:
In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some symmetries as motions, as opposed to discrete symmetry, e.g. reflection symmetry, which is invariant under a kind of flip from one state to another.

The notion of continuous symmetry has largely and successfully been formalised in the mathematical notions of topological group, Lie group and group action. For most practical purposes continuous symmetry is modeled by a group action of a topological group.

One-parameter subgroups
The simplest motions follow a one-parameter subgroup of a Lie group, such as the Euclidean group of three-dimensional space. For example translation parallel to the x-axis by u units, as u varies, is a one-parameter group of motions. Rotation around the z-axis is also a one-parameter group.

Noether’s theorem
Continuous symmetry has a basic role in Noether’s theorem in theoretical physics, in the derivation of conservation laws from symmetry principles, specifically for continuous symmetries. The search for continuous symmetries only intensified with the further developments of quantum field theory.

See also:

References:  William H. Barker, Roger Howe, Continuous Symmetry: from Euclid to Klein (2007)


Your Small Group For Controversial Discussions

The Nature of the Finite-Infinite Relation

December 2017 Update: April 2018 OVERVIEW  WORKSHOP Session #1  #2  #3 #4 #5 #6 #7 #8 #9  Summary

Convener-Presenter:  This could be you.

Purpose of this group. To develop a special graciousness and openness about life and beliefs, this group is for the people who want to get along with believers and non-believers. Can we find a deeper truth that holds up some new insights that will embrace both sides of an equation by focusing on the nature of the relation. We’ll engage the edges of scientific research and its implications for our current theories about the universe and its origins, and about who we are and the meaning and value of life.

Overview: One of the best sources for a study of the relation between the finite and infinite are the sacred texts within our religious traditions.  Many of us who grew up in the Christian tradition and like typical college students, we tend to reject the old to begin to self-actualize.  Then we begin studying at the limits and boundaries of our knowledge and the challenges can become so daunting, they get left in those undergraduate and graduate classrooms. Personally I went out as far as I could on the edges of physics, working with Bob Cohen, then chairman of Boston University’s physics department, and then with physicists from around the world. I began discovering there are conceptual overlaps between all the departments within the university. The most extreme appeared to be those who were religious and those who demonized all religions.

What can be more different that the texts within The Bible, both Old Testament and the New Testament, and most texts within the scientific community, that is between Genesis 1 and John 1 and Stephen Hawking’s big bang theory.  These sessions are designed to examine concepts within the sciences, philosophy, ethics, and mathematics to see how and where these overlap with concepts about eternity, infinity, light, and love. We start with an integrated view of the universe, and that begins to inform our understanding of the infinite and infinity.

Structure: This small group will have just nine gatherings, no longer than 90 minutes each. A goal for these sessions is to chart a way to empower people to create such a small group study.
• The first two sessions. We will re-examine cosmological models of the universe (2 weeks).
• The 3rd and 4th sessions:  We will explore various ways of approaching an understanding of the finite-infinite relation.
• The 5th and 6th sessions: We will explore a rather different understanding of light, a light that permeates and defines every notation and all of space and time.
• The 7th and 8th sessions.  We will explore the challenges to our understanding of basic concepts like space and time.  Both become finite and transaction oriented.
• The final session: Beyond the summaries, we will be searching for answers to the question, “What do we do now?”

Simple and small goals:  The first goal is to open the door to a very simple orientation to science and faith that (1) works with science and mathematics and (2) allows for, and possibly informs, religious beliefs. The next goal is to explore the entry points between the finite and infinite. Another goal is to explore the physics, philosophy, and psychology of light. If we have even limited success, we’ll all begin to shrink space and time and open up an intimacy with the universe.

Resources to start your own Study Group:

Promo sample to paste on bulletin boards: Please Update the dates, locations and people.
Update: PDF to be printed back-to-back after updating. It creates four 4.25×5.5 inch handouts.
Session 1Models of the Universe   Worksheet #1   S1a   S1b   S1c
Session 2: Models of the Universe
Session 3: Finite-Infinite: Continuity-Symmetry-Harmony
Session 4: Finite-Infinite: Order-Relations-Dynamics
Session 5: Light as defined by the Planck Length and Planck Time
Session 6: Light and time. Ten ways to reconsider the nature of time.
Session 7: Eight Key Ideas
Session 8: Planck Explains Einstein and Redefines Space-Time and Pi.
Session 9: “It’s a wrap.”
1. Always check the homepage of
2. Check for any prior homepages you may have missed.
3. Add to the dialogue: Comment, like, Tweet, Link


Upon following the work of science writer, Dan Falk

quantaDan Falk (on this website)
Author and journalist, Toronto, Canada
Homepage Wiki

Dan Falk is the author of  In Search of Time and a distinguished science writer from Toronto. His article, A Debate Over the Physics of Time,  was first published in  Quanta Magazine, July 19, 2016 and Atlantic Monthly, July 26, 2016 (just a week later).

Although most of the article is a focus on the June 2016 conference at the Perimeter Institute for Theoretical Physics in Waterloo, Ontario, there are many important concepts and references in this article that need to be part of one’s understanding of the the struggle to understand time.

All of these concepts will be studied in light of our simple model using simple concepts and simple math.

On a homepage here Go to the simple base-2 chart

Smolin, Lee

Lee Smolin


Founding member and Senior Faculty
Perimeter Institute for Theoretical Physics

Waterloo, Ontario, Canada

ArXiv   CV   Homepage    Twitter Wiki
P.S. An excellent little website, Crisis-in-Physics, once summarized and gave access to this professor’s articles. Unfortunately it has been shut down.

Fourth email: Thursday, December 30, 2020

Dear Prof. Dr. Lee Smolin:

You, Frank Wilczek, Paul Davies, and a few other leading scholars
are open enough to tell us why our simple construct (of the universe) just
might warrant further attention. It came out of a high school geometry class.

We did a simple thought experiment and went deeper and deeper within
the tetrahedron and its internal octahedron by dividing the edges in two
and connecting the new vertices. In 45 steps we were in the size range
of particle physics. In 67 more steps within we were studying the Planck
base units. We also doubled our original models; and, in just 90 steps, we
were out to the age and size of the universe. We created a chart with all
that very simple math:

It was intriguing because our cold start (like Lemaitre’s 1927 model) actually
compared favorably with an infinitely-hot start. Of course, our start had
a natural inflation and easily accommodated homogeneity and isotropy.

Shall we continue to pursue our simple model where base-2 notation
has been applied to the Planck base units? All notations are dynamic,
there is a perpetual start, the first 201 notations are symmetric, and that
arrow of time exists only in Notation-202. I think it has a lot going for it,
but it needs scholarly counsel and perspective. Thank you.

Most sincerely,

PS. Happy New Year! My hope is that everyone has a better year, but
I will admit, my optimism has been dented and tarnished… -BEC

Third email: Tuesday, October 4, 2016

Dear Prof. Dr. Lee Smolin:

Congratulations on all that you have done, especially for the Perimeter Institute. Phenomenal.

We are creating links to your work and the Perimeter Institute. As our journey progresses, there will be two pages about Perimeter —  — and a brief overview of the conference, Time in Cosmology ( )

There is also a general overview page of your work:

If there is anything you’d like changed, please just say the word!


* * * *
Bruce Camber

PS. Yes, I know how naive and idiosyncratic our work is. The simplicity of the logic and math, however, has caught our attention. The numbers seem to speak louder than words. Although temperature is a problem, I think in time we’ll be able to adjust that line of figures with some kind of “reasonable” rationale. -B

Our second email: July 25, 2016

Dear Prof. Dr. Lee Smolin:

I swear the years are becoming superluminal they’re going so fast.
It’s the Inflationary Epoch all over again.

In 2011 a group of high school folks (teachers and students) began mapping the universe using base-2 exponential notation from the Planck base units to the Age of the Universe. We fell into a tetrahedron and kept sliding to the center…

We tried using The Trouble with Physics as our rappelling ropes, but around the 67th notation, down with the fermions and protons, those ropes quickly turned to strings so we dropped into a virtual free fall until squeezed at the door of “the singularity” with Max’s secret codes. Wilczek gave us some clues on interpreting the codes. We just got lucky and found our way out and then went up the next 90 notations to the Now.  Quite a trip. Just over 200 notations! 65 or so had never been explored! Incredible, isn’t it? Just a silly daydream?  Could there be anything to it?

So, we’ve been at it now for five years and eight months.
It’s time to get real or get serious. Can you help us?

All criticism is highly valued and encouraged!

Most sincerely,

First email:  20 September 2010
New Orleans


Dear Prof. Dr. Lee Smolin:

Your work has fascinated me over the years. You have always been larger than life. But now, we are getting older and genius seems to be more approachable with the web.

You know the tetrahedron. You know the octahedron. The quick question: What is perfectly enclosed within the octahedron? If you half the edges and push an octahedron in each corner, you’ll have a start. I wonder if you quickly knew the answer to that very simple, basic question about structure.

Most of the folks I have asked since friend and colleague David Bohm died in 1992 pull a blank. John Conway, and some of Bucky and Arthur Loeb’s folks figure it out or just know (back in the ‘70s I was part of Loeb’s Philomorphs).

I am looking in on your work that is posted on the web and then I will dig even deeper. I thought you might enjoy the simple question(please let me know if you knew the answer). I suspect you do not think it really matters. But wait, maybe it does…. Thank you.


Bruce Camber

PS. If you have a moment and you want to know more about why I think it does:
This is what I said to Len Mlodinow, Stephen Hawking’s collaborator (and the background story about Bohm):

We all need a summary statement about life and its meaning and value:

A little start on a TOE:

Of course, we should try to foment some anxiety for all those who have all the answers:

On following the work of Richard A. Muller…

Richard A. Muller, Professor of Physics, University of California – Berkeley
Senior Scientist at the Lawrence Berkeley Laboratory

ArXiv  Blog  CV  Homepage (Newer) Quora   Wiki    YouTube

Key references  (books):
NOW: The Physics of Time, September 2016 (Amazon)
There is no past and future, only this time, right now.
> He holds onto big bang cosmology.
> He throws entropy under the bus (Now, Chapters 9, 10, 15).
> He likes infinity. Infinity works.
> He continues the Cartesian logic, Cogito Ergo Sum.

References to Richard Muller’s work within this website: Time Is Always Right Now With Carlo Rovelli questioning our understanding of time.

Third email: 5 November 2019 @ 8 AM

Dear Prof. Dr. Richard Muller:

All references to you within our website are with great respect and deference.
This page:

The most recent reference is here:
It is also the current homepage.

I hope you are well and your work rewarding.
Most sincerely,

Second Email:  Wednesday, 8 March 2017

Dear Prof. Dr. Richard  Muller,

As part of a blog I have probably bastardized your intentions with Now: The Physics of Time.  I’ll insert part of that blog just below.  Also, I am slowly developing this page about your scholarship.

I am not a scholar, just a simple idiosyncratic thinker.

If you would have me delete “your” page, I will. If you would like me to revise it, I will. The site and all the writings are an attempt to explain the simple math and logic that follows base-2 exponentiation from the Planck scale to the Age of the Universe. In looking at those numbers, a few years ago, I began to doubt the logic of big bang cosmology.  When I noticed that the numbers nicely mimicked the epochs defined by big bang cosmology, I thought a nice, organic, natural inflation that was more predictive and simple than the old Lemaitre theory and GuthLinde‘s inflation, might have some merit.

I apologize for taking your time this way and thank you in advance for any considerations of these topics.

Most sincerely,

Bruce Camber

PS.  I said the following about you and your work:
Today-Now versus Commonsense and Time’s Past-Present-and-Future

“Just because we have all believed the same thing for over 300 years doesn’t make it true. Yes, “commonsense worldviews” are the hardest to breakdown. Cognitive dissonance holds the line.

“This is the new reality. There is no yesterday or tomorrow.  There is no past and future. There is only Now, today, this moment. In 1715 Sir Isaac Newton got us all off on the wrong foot. He made space and time absolutes that define infinity. The only problem is space and time are derivative, discrete, finite, local, quantized, and relational.

“If logic dictates, Prof Dr. Richard A Muller is right. This Berkeley physics professor boldly claims that the only time is Now.  And, he subtitled this book, The Physics of Time.  There is no past, no future, everything for all time is always present. The entire universe is a process and our only reality. The first moment of creation is Notation 1 and it is always a dynamic part of the very beingness of our universe. It is not past; it is happening right Now, sustaining and creating, the initial step of our universe.

“Everything you do is instantiated within the very being of the universe and becomes part of its evolving definition.”

First Email:  26 September 2016

A warm “Congratulations” for NOW.

Might our simple integrated math and logic apply?

Our little history:

Most sincerely,
PS. I see you grew up in the nice part of the Bronx! I taught 2nd grade in 1969 in PS 48x in Hunts Point, the Bronx.  Those were crazy times!  -B

Upon discovering the work of Roberto Mangabeira Unger…

Roberto Mangabeira Unger, Harvard Law School, Cambridge, Massachusetts


References within this website:

Second email:  Tuesday, 4 October 2016

RE: Nihilism and the lack of an integrated universe view

Dear Prof. Dr. Roberto Unger:

Thank you for your work on  Beyond Nihilism: Directions in the Spiritual History of Mankind.

In December 2011 we “fell into” an integrated view of the universe that started with the Planck base units and went to the Age of the Universe in just over 200 base-2 notations.     ( )

We believe that was a first for a base-2 progression. It is simple. It is mathematically integrated. And, it just may be a clear alternative to the big bang (and the big bang nihilism).  Most of the people at your conference, Time in Cosmology, at the Perimeter Institute accept the place of the big bang as a given.

To help our students and to attempt to context that diverse dialogue, I have created a few links to the conference and to your work.  There currently are three key pages. First, there was a brief overview of the conference on our homepage, and now,

There is also this page on the conference:

Our general overview page to your work is here:

If there is anything you’d like changed, please just say the word! Thanks.

Now, thinking about time and the large-scale universe, perhaps another conference could be entertained, Time in the small-scale and human scale universe. In less than a second, the universe within this base-2 model has already expanded well in to the large-scale universe. Of the 200 notations, the first second from Planck Time is within notations 144-145. The first day (86400 seconds) is between notations 160 and 161. A light year is between notation 168 and 169. That’s cosmology.

If we engage the numbers generated using base-2 from the Planck base units, it all appears to expand rather quietly right out beyond the need for a big bang.


* * * *
Bruce Camber
New Orleans

PS. Yes, I know how naive and idiosyncratic our work is. The simplicity of the logic and math, however, has caught our attention. The numbers seem to speak louder than words. Although temperature is a problem, I think in time we’ll be able to adjust that line of figures with some kind of “reasonable” rationale, perhaps a different algorithm. -B


First email: January 1, 2015

Roberto Mangabeira Unger
Roscoe Pound Professor of Law

Dear Prof. Dr. Roberto Unger:

I read with great interest your recent work with Lee Smolin.  You may
find two of our very unusual conclusions to be of some interest.

From our work within a high school in New Orleans, we have begun to
believe that there is an ethical bias deep within the very fabric of
the universe:

Also, regarding time, we ask, “Might time simply be a sense of time?”
Within our work time appears to be derivative.  Planck Time provides
the beginning and the Age of the Universe provides the end of a very
finite scale.  There are just 202.34 base-2 exponential notations from
beginning to end; and as you might anticipate, the Planck Length and
Planck Time track well together.  Though I had not see this chart
until we created it, you might find it to be of some interest.

When just a kid in the ’50s, I grew up playing childhood games
throughout the areas of the Law School, Jefferson Lab and the Peabody.
So much has changed!

Best wishes for the New Year.

Most sincerely,

Bruce Camber