Arcadian Functor

occasional meanderings in physics' brave new world

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Marni D. Sheppeard

Friday, July 27, 2007

Higgs Blogging II

The Higgs mechanism is without a doubt one of the most beautiful features of the Standard Model. This may, however, be an argument in favour of considering it a natural effective mass generator, rather than an argument for taking its particle nature seriously. Mathematicians such as N. Hitchin have run with Higgs fields and constructed fantastic spaces. Alain Connes has even reformulated the Standard Model using NCG. But again, this seems to be an argument in favour of the Higgs being a stopgap for something deeper. For those of us who work on a QFT not based on symmetry or Lagrangians, the Higgs can only potentially exist as a derived concept, in which case there is no need to posit it a priori.

But the biggest problem with the Higgs is its attachment to the vacuum, and its pervasion of space. That's simply not something that particles do. So just because nobody has yet fully explained the particle masses any other way, doesn't mean the Higgs boson exists. How can we possibly hope to understand the Higgs mechanism without understanding mass?

A guest poster on The Everything Seminar today points out that most of our baryonic mass comes from the binding of quarks and other partons in atomic nuclei, and not from the rest mass of quarks themselves. In other words, an unseen seething relativistic swamp of matter, for which we don't even have a real theory, makes up most of the mass we care about, unless we want to consider Dark Matter as well, in which case we clearly don't know very much about anything. This is the realm of quantum gravity, not the Standard Model. Assuming Dark Matter requires a whole new category of observables, obliterating our trust in the old ones, and leads to the expectation of new physics at the TeV scale.


Blogger Mahndisa S. Rigmaiden said...

07 26 07

Hey there Kea:
Thanks for commenting on my dog neutering article. Your post was excellent! I first heard of the Higgs years ago while attending a lecture by Mary K. Guillard. She was awesome! However, what struck me the most was that the Michelson Morely experiment showed there was no ether, but in principle, the Higgs field pervading all of space sounds a bit etherish.

Once I thought a bit, I figured there had to be other underlying mechanisms that generated massivity of particles. What is it? I am unsure, but if the Higgs particle is really out there, LHC will reveal it methinks.

One might posit that quark binding creates particle massivity, but what gives the quarks their masses? It comes down to a Russian doll metaphor of sorts...OK I will leave now, offline life is busy as all hell right now!

Have a great rest of week!

July 27, 2007 4:38 PM  
Blogger Kea said...

Mahndisa! Good to hear from you. Yes, if the Higgs exists the LHC should really see it, but it will take longer to dig out of the data than other exciting possibilities like 'gravitino' type dark matter.

July 27, 2007 4:45 PM  
Anonymous Anonymous said...

Kea, you said "... the biggest problem with the Higgs is its attachment to the vacuum, and its pervasion of space. That's simply not something that particles do ...".

What do you think of the possibility that the Higgs is really a condensate of T quarks and antiquarks ?

Such models have been described in detail by Yamawaki et al, and the results of their calculations seem to be consistent with experimental results.
My discussion of Triviality and Vacuum Stability at
has references to the work of Yamawakli et al.

It seems to me that their work indicates that the Higgs need not be added ad hoc to the Standard Model, but can be a sort of global condensate along the lines that it seems to me you are suggesting,
and that the physical result may be something that could be detected at LHC,
so even detection at LHC of a Higg-like T-TBar thing might ber consistent with your views.

Tony Smith

PS - I am trying to work out the relationships between Spin(2,4) with conformal Dark Energy and Spin(3,3) with varying c, and if I can figure out something useful I will try to post it on your blog as a comment.

July 27, 2007 4:54 PM  
Blogger Matti Pitkanen said...

It seems that there is a renessessaince in thinking about particle massivation. Great if theoretical particle physics could be resuscitated.

I wrote my own comments but they are so lengty that I don't want to bore you here and do this at my blog.

July 27, 2007 6:17 PM  
Blogger nige said...

"However, what struck me the most was that the Michelson Morely experiment showed there was no ether, but in principle, the Higgs field pervading all of space sounds a bit etherish." - Mahndisa

The Michelson Morley experiment have a null result for absolute speed of light carried by "aether".

FitzGerald and Lorentz made that null result consistent with "aether" by postulating a contraction of the instrument, so that light travelling along both paths in the Michelson Morley experiment does it in the same time (the contracted distance offsets a varying speed of light, according to FitzGerald 1887 and Lorentz 1893).

So M. M. is not a disproof of the aether; it is a reason for making ad hoc modifications to the laws of physics (introducing the FitzGerald-Lorentz transformation). This isn't speculative, because Lorentz found from this (years before Einstein's relativity) that you get falsifiable predictions associated with contracting J.J. Thomson's electron: mass increases with velocity.

The "aether" failed because it wasn't a single model by a landscape of 200 models, so it wasn't falsifiable ("disprove" one model and there are lots of others, or the innovator of the "disproved" model would just add another idler wheel system to "fix" the error).

Some of the "aether" theories were dismissed by Maxwell (an "aether" theorist himself) for false reasons: he claimed that exchange radiations can't cause forces because they would heat up bodies.

Actually, if Maxwell's argument was correct it would now be debunking the Standard Model of particle physics, which is based on exchange radiation. Maxwell's error is assuming that all radiation energy must be degraded into heat energy: gauge bosons are known to not have this property. They are different in nature to "real" radiation, and they only produce forces, not heat. They don't make charges oscillate at a frequency which causes them to radiate; just push and shove smoothly, causing forces.

So the main problem with "aether" is the issue of drag forces.

It was argued repeatedly by many people that any particulate field in the vacuum would cause planets to slow down and spiral into the sun, like air resistance.

Unfortunately for such critics, when a gas molecule hits the front of a moving car and bounces off to cause the drag effect, the molecule must end up with more energy than before, so that kinetic energy of the car gets converted into kinetic energy (and heat) of the air.

Otherwise, the car can't be slowed down: so for drag to work there must be a conversion, somehow, of kinetic energy from the moving car to the particles in the surrounding gas or "aether".

This is not always possible unless the speed of the particles can be speeded up, and of course particles going at the velocity of light can't get speeded up by impacts, so they can't take energy out of a collision (unless their frequency is altered).

In addition, bosons don't always interact with one another, so they don't degrade any gained energy as heat by hitting one another (bosons only interact with one another if they are charged, e.g., the weak effect of gravity, whereby all energy is a source of curvature according to GR).

This is because bosons don't obey the exclusion principle, so they can pass through one another and emerge unscattered.

So bosons don't automatically dissipate kinetic energy from a moving body and slow things down. They don't cause drag, just effects like contraction and inertial mass increase (the snowplow effect).

Kea: I'm sorry if this is too long and unhelpful, please delete if it is. I'll keep a copy on my blog.

July 28, 2007 2:24 AM  
Anonymous Anonymous said...

Kea, if this comment is too long or off-topic,
please feel free to delete or move it.

Danny Ross Lunsford, in his paper on the CERN preprint server at EXT-2003-090 dated 28 November 2003 and entitled "Gravitation and Electrodynamics Over SO(3,3)" said:
"... we obtain general relativity only in the limit

/\ = 0 , R [goes to] 0 , W [goes to] 0 , R/W [goes to] - 4 pi G ...

the ... cosmological constant ... is an artifact of the decoupling of gravity and electromagnetism ... the Einstein-Maxwell equations are to be regarded as a first-order approximation to the full calibration-invariant system ... the cosmological constant must vanish in order that this limit exist ...".

So, it appears to me that Dark Energy (the cosmological constant) would also appear using signature (3,3) if you allow DRL's electromagnetic part to fully interact with the gravitational part (as I do in my model, in which I interpret DRL's electromagnetic part as conformal graviphotons
rather than electromagnetic photons),
if you do as DRL and get the Einstein-Maxwell equations, you need to restrict to zero Dark Energy / cosmological constant,
that Louise could indeed use Step 11 of my construction to get a realistic varying-c model by using the same criterion that DRL used to get the Einstein-Maxwell equations, using either signature (3,3) or (2,4).

Although he did not cite the 2003 work of Danny Ross Lunsford, George A. J. Sparling, in gr-qc/0610068, said:
"... a new spinorial theory of physics is developed ...[it]... rounds out the "primordial theory" of the author and Phillip Tillman [cond-mat/0401015] which supposes there is a triality symmetry ... associated with ... O(4,4) ...
triality requires that space-time extends minimally to six-dimensions, of signature (3,3), so it predicts two extra timelike dimensions
The triality spaces have conformal symmetry group ... O(4,4). We need to reduce this to O(2,4) to give the conformal symmetries of Minkowsi space-time ... for the two twistor spaces, we need to reduce to U(2,2) ...
This structure extends naturally and beautifully to six-dimensional space-time ... of signature (3,3)
the excitations at the boundary ...[of]... the spaces arising at the boundaries of the ... fundamental quantum fermionic fluid of Shou-Cheng Zhang and Jiangping Hu ... giving rise to the structure of the spaces. ...".

Sparling's construction that explicitly went
O(4,4) to O(2,4) to U(2,2) to signature (3,3)
seems that to confirm my view that the Spin(2,4) conformal stuff is physically equivalent to the Spin(3,3) stuff.

As to the "primordial theory" of Sparling and Tillman at cond-mat/0401015, there they say:
"... We discuss a series of possible transitions, taking us from a real form of E8 down to the group SU(2,2) and to the Poincare group.
... We shold note that something like this scenario has occurred to others beforehand, but here the physics involved is apparently different [43].
[43] T. Smith, WWWHome page, ...".

It is nice that they mentioned my work,
their reference is to a web site that is no longer functional,
because their paper was written in January 2004 and it was not until December 2004 that my current web site was set up,
since I had been blacklisted from the Cornell arXiv since 2002, there were no arXiv references to my then-recent work.

Perhaps Sparling's failure to cite the work of Danny Ross Lunsford may have been due to him also being blacklisted by the Cornell arXiv.

Had we not been blacklisted, maybe we could have had some discussions with Sparling and/or Tillman, which might have resulted in a more rapid
advance of physics, and a better understanding of each other's work.
For instance,
I do not agree with Sparling and Tillman that the physics of my scenario is "apparently different" from their scenario of transitions.

Here is a comparison:

Sparling E8
Smith E8

Sparling E7
Smith E7

Sparling E6
Smith E6

Sparling Spin(5,5)
Smith Spin(10) (various signatures)

Sparling Spin(4,4)
Smith Spin(8) (various signatures)

Sparling SU(2,2)
Smith Spin(2,4) = SU(2,2)

Sparling Poincare
Smith Poincare

What might be differences?

At the E6 level, I get the Standard Model from a string theory with strings being interpreted as world-lines (the natural 1-dim things in spacetime), which is what leads me to the Monster Group as symmetry of a single cell of lattice spacetime.

At the Spin(8) level, I use Clifford algebras and 8-fold periodicity to construct a generalized Hyperfinite II1 von Neumann factor to use with respect to algebraic approaches to physics.

I use the parts of Spin(8) left over from the transition to SU(2,2) to get the SU(3)xSU(2)xU(1) of the Standard Model,
in doing so get a M4xCP2 spacetime like that of Matti Pitkanen (actually my use of M4xCP2 was based on discussions with Matti).

My M4xCP2 8-dim structure is used to construct a Higgs-Without-Higgs model based on the Higgs being a condensate of T quarks and antiquarks,
which is similar to Sparling's use of Zhang-Hu "fundamental quantum fermionic fluid".

As to Sparling's "excitations at the boundary ... giving rise to the structure of the spaces ...",
in my model the boundaries are Shilov boundaries of complex domains, so that the boundary "excitations" do indeed "giv[e] rise to the structure
of the spaces", and the volumes of those geometric structures (along with some combinatorics) allow me to calculate particle masses, force strengths, etc that are substantially consistent with experimental results.

At the SU(2,2) level, I used conformal structures instead of twistors, but, as Penrose and Rindler wrote in their book, those two approaches are equivalent.

At the SU(2,2) level, I could have used Danny Ross Lunsford's criteria to get a model like Louise's with varying c and zero Dark Energy, something that Nige has carefully shown to be reasonable physics.
I did not do that.
Maybe I am wrong and Louise could be right and do that.

It seems to me that the possible differences are not really differences between Sparling's model and my model, but are mostly areas where I have
gone further than Sparling, and it is likely that he (maybe with help re Clifford algebras from Carl Brannen and/or Garrett Lisi, and from Matti Pitkanen with respect to M4xCP2) will soon get there also.

Louise said in a comment on another thread in this blog:
"... I hope we don't grow up to be old and angry ...".

As for myself, I am already old (66 this year) and I fear that seeing the consequences of my work being blacklisted by arXiv does make me angry.

I wish that I were a good enough person to just be happy to see major elements of my work being rediscovered by others,
I so much regret not being invited to the rediscovery party
that I fear that bitterness, and maybe even anger, may be my dominant feeling.

Tony Smith

PS - With respect to claims that Enzo Bonacci has priority with respect to "tridimensioinal time", it seems to me that his publications in 2006 are not as old as the work of Danny Ross Lunsford, which was on the CERN preprint server on 28 November 2003.
I am NOT trying to attack the work of Bonacci, as it is in my opinion a sign of high talent and good work to rediscover something interesting, but am only saying that as far as I know, it seems to me that priority in time is with Danny Ross Lunsford.

July 28, 2007 6:58 AM  
Blogger Kea said...

Hi all. Tony, I really appreciate your comments. Yes, I think the tt condensate is an excellent idea, and I would like to spend some time looking through your work and links to Yamawakli et al. as soon as I get the chance.

As for bitterness, I must say I cannot imagine how someone in your position could escape this feeling, after years and years of being basically ignored, even when the ideas begin to be picked up by other people and published. Probably we'll all end up bitter and twisted. That thought, however, only makes me more angry and determined to continue on this tack.

July 28, 2007 10:51 AM  
Blogger kneemo said...

Hi everybody

This sunday I plan to attend a talk by Itzhak Bars at UCLA on his two-time physics theory, S-theory. S-theory grew out of investigations of SO(10,2) covariant supergravity, with real 32-dimensional Weyl spinors.

July 28, 2007 1:53 PM  
Blogger Kea said...

Excellent, kneemo. Please tell us how it goes.

July 28, 2007 2:17 PM  
Blogger kneemo said...


S-theory's algebraic structure looks very similar to the 5-grading for E7(-25) where g(0)=so(2,10)+R, dim(g(-1))=32 and dim(g(-2))=1. I've been wondering if there is some three-time theory with structure similar to the 5-grading of E8(-24) with g(0)=so(3,11)+R, dim(g(-1))=64 and dim(g(-2))=14. Perhaps there would be a related SO(3,11) covariant supergravity in this case. Have you looked into this any?

July 28, 2007 2:49 PM  
Blogger Matti Pitkanen said...

The censorship in archive is extremely cruel form of discrimination and should be treated as a crime against basic human rights. Also as a victim of it I must confess that bitter feelings are unavoidable now and then. My own reaction has been to devote the rest of life to articulate my lifework as precisely as I ever can without wasting time to attempts to publish in so called respected journals. Kind of fanatism but saves my mental health. The positive side is that although they can label us crackpots but they cannot destroy the documentations at our home-pages.

I had years ago very useful discussions with Tony. In particular, I learned a lot about octonions and quaternions from Tony and started seriously to consider the possibility that dimension 4 and 8 might really relate to them somehow although the metric signatures seemed to be an obstacle.

July 28, 2007 3:30 PM  
Anonymous Anonymous said...

kneemo said:
"... S-theory's algebraic structure looks very similar to the 5-grading forE7(-25) where
g(0) = so(2,10)+R, dim(g(-1))=32 and dim(g(-2))=1.
I've been wondering if there is some three-time theory with structure similar to the 5-grading of E8(-24) with
g(0) = so(3,11)+R, dim(g(-1))=64 and dim(g(-2))=14

I have not looked into that (thanks very much for mentioning the idea), but it is very suggestive to think that the following gradings might correspond respectively to 1-time, 2-time, and 3-time:

E6(-26) where
g(0) = so(1,9)+R, dim(g(-1))=16

E7(-25) where
g(0) = so(2,10)+R, dim(g(-1))=32 and dim(g(-2))=1

E8(-24) where
g(0) = so(3,11)+R, dim(g(-1))=64 and dim(g(-2))=14

Maybe Itzhak Bars might comment on that in his UCLA talk tomorrow?
If he does, could you (kneemo) post a comment here saying what he said?

Tony Smith

July 29, 2007 2:08 PM  
Anonymous Anonymous said...

I said in an earlier post that it seemed to me that the following gradings might correspond respectively to 1-time, 2-time, and 3-time:

E6(-26) where
g(0) = so(1,9)+R, dim(g(-1))=16

E7(-25) where
g(0) = so(2,10)+R, dim(g(-1))=32 and dim(g(-2))=1

E8(-24) where
g(0) = so(3,11)+R, dim(g(-1))=64 and dim(g(-2))=14

The 3-time E8 structure can also give a 4-time structure if you just look at it in terms of even and odd grades.

Consider the even part g(ev) of the 3-time grading E8(-24) where
g(0) = so(3,11)+R, dim(g(-1))=64 and dim(g(-2))=14

g(ev) = g(-2) + g(0) + g(2)
so that
dim(g(ev)) = 14 + 91+1 + 14 = 120 = dim (so(4,12))

and we have

g = E8(-24)
g(ev) = so(4,12) which has a 4-time signature with the odd graded part g(-1) being 64-dimensional.

See Table 8 of Soji Kaneyuki's chapter entitled Graded Lie Algebras, Related Geometric Structures, and Pseudo-hermitian Symmetric Spaces,
as Part II of the book Analysis and Geometry on Complex Homogeneous Domains,
by Jacques Faraut, Soji Kaneyuki, Adam Koranyi, Qi-keng Lu, and Guy Roos (Birkhauser 2000).

Therefore, the 1-time, 2-time, 3-time, and 4-time structures might look like:

E6(-26) where
g(0) = so(1,9)+R, dim(g(-1))=16

E7(-25) where
g(0) = so(2,10)+R, dim(g(-1))=32 and dim(g(-2))=1

E8(-24) where
g(0) = so(3,11)+R, dim(g(-1))=64 and dim(g(-2))=14

E8(-24) where
g(ev) = so(4,12), dim(g(-1))=64

Tony Smith

July 30, 2007 9:43 AM  
Blogger kneemo said...

Hello everybody

The two-time physics talk by Itzhak Bars was quite enjoyable. The content of the talk consisted of the SO(4,2) compactification of S-theory. Some related slides can be found online here.

About halfway through the talk, Bars noted that adding more than two extra time dimensions is troublesome. When I asked about this later, he clarified this point, stating that one might indeed be able add more time dimensions as long as one can find a clever way to get rid of the ghosts (negative norm states). Bars said he is only able to do this in the two-time framework by invoking a position-momentum symmetry. For a three-time theory, such as an SO(11,3) supergravity, he said one would need to find a new symmetry to eliminate the ghosts.

July 30, 2007 6:05 PM  
Anonymous Anonymous said...

kneemo, thanks for your report on the talk of Itzhak Bars, who mentioned possible objections to more than two times in physics.

As to more times, Paul Bird has written about 4-time physics in 16-dim in:

Years ago, I had a brief e-mail correspondence with Itzhak Bars.

Back in 1999 (before the arXiv went to Cornell and I was blacklisted) I wrote hep-th/9908205 in which I suggested physics in
"... 6 spacetime dimensions with local Conformal symmetry of the Conformal Group C(1,3) = Spin(2,4) = SU(2,2) ...".

In February 2000, I wrote to Itzhak Bars asking him what he thought about "... decompos[ing] a 10-dimensional spacetime into
a 6-dimensional physical/conformal spacetime
a 4-dimensional compactified internal symmetry space ...".


Itzhak Bars replied (also in Febraury 2000), saying in part:
"... Six dimensions is not essential ...".

Then, in August 2000, Bars wrote hep-th/0008164 saying:
"... The Standard Model of particle physics can be regarded as a gauge fixed form of a 2T theory in 4+2 dimensions. ...".

Now, the web page of Itzhak Bars says:
"... Amazingly, the best understood fundamental theory in Physics, the Standard Model of Particles and Forces (SM) in 3+1 dimensions, is reproduced as one of the “shadows” of a parent field theory in 4+2 dimensions. But even more amazing is that this emergent SM has better features than the ordinary SM in 1T-physics. ...
The permitted motions in 4+2 phase space are highly symmetrical, as they are constrained by a Sp(2,R) gauge symmetry that makes momentum and position indistinguishable at any instant. ....".

The symplectic structure of Sp(2) gives the momentum-position duality, and
the isomorphism Sp(2) = Spin(2,3) gives the inclusions

Lorentz Spin(1,3) inside Spin(2,3) inside Conformal Spin(2,4)

so that the Sp(2) = Spin(2,3) momentum-position duality fits naturally.

Something that Itzhak Bars does not (AFAIK) yet do is to get gravity by a version of the MacDowell-Mansouri mechanism (described by Mohapatra in section 14.6 of his book Unification and Supersymmetry (2nd edition, Springer-Verlag 1992)) which also uses the inclusions

Lorentz Spin(1,3) inside Spin(2,3) inside Conformal Spin(2,4)

in which the Spin(2,3) = Sp(2) is used as the antideSitter group.
I have worked on that approach, and it gives a prediction of the Dark Energy :Dark Matter : Ordinary Matter ratio of 75.3 : 20.2 : 4.5
Dark Energy comes from the special conformal generators of the Conformal Group Spin(2,4)
Dark Matter is primordial black holes.

In short summary, I think that the work of Itzhak Bars is very interesting,
I am happy to see advances in the physics of 4+2 conformal spacetime
I am not sure about his objections to the 4-time physics of Paul Bird.
For instance, there might be some dualities among branes etc that could be used as Itzhak Bars uses Sp(2) symplectic duality.

Tony Smith

July 31, 2007 8:08 AM  
Blogger Kea said...

....stating that one might indeed be able add more time dimensions as long as one can find a clever way to get rid of the ghosts (negative norm states)

Did you tell him about operads, then?

July 31, 2007 3:16 PM  
Blogger kneemo said...

Did you tell him about operads, then?

Nope, I didn't get a chance to get there. The non-compact exceptional groups and their relations with S-theory and some mysterious three time extension ate most of the time up. I didn't get to talk to him very long because his friends were taking him to dinner.

July 31, 2007 6:37 PM  
Anonymous Anonymous said...

Hello Kea,

if you give us the opportunity, we try to explain the reasons why Enzo Bonacci should have the priority on tridimensional time theory.

WE belong to the “maths’ friend circle”, people around the world who desire to support our shy friend Enzo Bonacci to get the notoriety he deserves. If he gets famous you will know our identities as well.

ENZO BONACCI is the greatest mind we know, able to speak ten languages, to play chess with different people toghether, consultant for strategic defense in the past (now convinced pacifist) and for several boards today, indefatigable teacher and writer. His greatest defect (in this imageaholic society) is shiness. That’s why he needs friends. After years striving, the Italian Scientific Community has decided to accept his theories about Relativity. You only know two parts of a trilogy of Relativity revision because the consequences in terms of energy sources will be huge and he fears military applications.

GEORGE SPARLING is a former Phd pupil of Roger Penrose, the first physicist to whom Enzo Bonacci sent his publications. The coincidence of his claiming for a new esadimensional theory some months after, seemed rather strange to our eyes, but in the meantime Enzo and George have become friends because their works could merge without collision and they are both reasonable people.

RAYMOND CRITCHLEY is the real father of esadimensionality (1978) but in different terms from Bonacci and Sparling.

DANNY ROSS LUNSFORD is a brilliant researcher whose work enforces the esadimensional theory in a way different from both Bonacci’s diode-photodiode ideal experiments or Sparling’s spinorial calculations, but somehow close to Critchley’s trace anomaly explanation.

INDIRECT PROOFS: like the ones provided by Critchley, Sparling and Lunsford are marvellous analytic essays, but according to our opinion less important than the direct measures by Enzo Bonacci. He actually measured time in different positions, having the courage to call the extra-dimensions “time” and not in exotic ways as Critchley’s “integrated dimensions”, Lunsford’s “coordinatized matter” or Sparling’s “time-like dimensions”.

TRIDIMENSIONAL TIME is an expression you couldn’t find on research engines until 2006 January, when Enzo Bonacci published on-line his work reaching soon ranks of 100000 people visiting his site! By the way, copyright about his works (especially the plans for matter-antimatter and gravity engines) were registered some years before... He just delayed the publications frightened of possible weapons until the time he realized that oil-wars were frightening as well.

FAIR PLAY has always characterized Enzo Bonacci who is going to quote ALL past essays in the long way to the esadimensionality (From Kerr esasolutions of black holes to Sparling esaXitransform) in an official way so that all the people who have worked about this problem could have their deserved place in physics and it will be the Community to judge their contributions. We invite all the other pioneers in esadimensional field to do the same.

Thank you all for your kind attention.


August 15, 2007 8:02 PM  

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