Top of pg 39, If I read correctly, you are stating the apparent angular size of the Sun is about 1 degree across. If that were true, there would be more frequent lunar eclipses, but no total solar eclipses. I believe the value is about half that value, around 0.5 degrees or 32 arc minutes, approximately the same size as the disk of the moon.

Also you need to be more careful with your use of onclick and urchinTracker

(I have a strange problem using chrome: each time I try to comment I’m redirected to Petrosky’s web page. It’s seems ok using explorer.)

Thx for your answer, although I’m not sure to understand. You seem to say that what happens in a flat matter-dominated universe has nothing to do with inflation. Then I was unclear: my question is about whether the “to any observer the size of the observable universe is the size of a black hole”? assertion may also hold in a universe with a big vacuum energy, or cosmological constant.

“the universe isn’t a black hole; if anything, it’s a white hole.”
What if white holes were black holes seen from the inside?

you confirmed my suspicion that you confound the ket |Psi> in the general N-particle case, with the special case described by wavefunctions as Y(x,t). Nowhere in the absoluteastronomy link that you gave above appears the word “field”, even once!

As explained to you before, it is only the last special case of Y(x,t), which can be interpreted as a field and next quantized using the formalism of second quantization.

You are right that “In any general sense, since real numbers are a subset of complex numbers, I can most certainly tell you that any number you produce has a complex number representation.” But this is a straw-man. Evidently the complex extensions of the Schrödinger equation used to explain some of the phenomenology of irreversible systems and of the arrow of time are those where purely real observables are extended by adding a non-zero imaginary part. If you read my messages with care, you would find the part where I gave the dissipation condition. I repeat it now: “In standard literature, the dissipativity condition is then defined as Im{H} =< 0″. When the imaginary part is zero, dissipation is zero and one recovers time symmetry.

I fail again to follow parts of your message. You did not reply to my questions for clarifications and it seems that you are rejecting the well-known fact that the Schrödinger equation is Markovian with your “since you don’t understand the general use of the word Markovian”.

The Schrödinger equation is valid only as approximation, when one ignores non-Markovian corrections, mixed states, random terms f… In the more general cases we use more general equations: from simple Ito-Schrödinger equations to more developed expression as Lindblad equation, Eu equation, the Brussels-Austin equation, etc.

An introduction to the Lindblad equation is given in the above Wikipedia link linked in my previous message.

I want just to add that the Brussels-Austin equation is based in a complex extension of the Liouville space and the condition for dissipativity is Im{Z} =< 0, where Z is an eigenvalue of the Liouvillian in a generalized space (beyond the Hilbert space of ordinary quantum mechanics). The applications to instable systems: particles, fields, etc. are found in standard literature in mainstream journals

http://order.ph.utexas.edu/people/Petrosky.htm

I did some remarks about how complex extensions of the Schrödinger equation can explain some of the phenomenology of irreversible systems, such as the decay of instable particles radioactive nuclei…

Let me do some other minor comments to your last message. First the Schrödinger equation does not need to be manipulating a wave function as you affirm; it can be manipulating a Ket or a Bra in Dirac’s abstract notation. A Ket |Psi> in the general N-particle case, cannot be represented using a complex number field. I think that you confound the general quantum state with the special case described by wavefunctions as Y(x,t).

I do not know what you mean by “a backward moving universe as being made of anti-particles”. It seems that you mean Stuckelberg-Feynman interpretation of antiparticles. Their model is not valid and in modern QFT, there is nothing like particles moving backward in time. There was a rather large discussion about this in newsgroup sci.physics.research the last two weeks.

I have no idea about what you mean by your “The Markovian processes with memory do represent our real universe”. Moreover, it is not true that “the schrodinger equation is what governs the probability evolution between each coin flip”. The equation is only valid in the Markovian limit, pure state, no random terms f… In more general cases we use more general cases: from simple Ito-Schrödinger equations to more developed expression as Lindblad equation, Eu equation, the Brushels-Austin equation, etc.

http://en.wikipedia.org/wiki/Lindblad_equation

Sorry but the scenario described in the section cited above is not assuming the second law of thermodynamics.

Same for the future. There will never be a future time that is infinite into the future from present time. The proof is very simple.

But the universe isn’t a black hole; if anything, it’s a white hole. Black holes have singularities in the future, white holes have them in the past.

Congrats on your book. If I may ask a question: can you (dis)prove “to any observer the size of the observable universe is the size of a black hole”?

I have been told this is trivially true for a flat universe. Should inflation changes this? What would be the consequences to take this as a law?

Best

The discussion at

http://www.canonicalscience.org/research/time.html

assumes the 2nd law of thermodynamics, which is not fundamental. Sean is postulating a scenario where the second law isn’t true before a certain time. Read the last chapter.

The time symmetry of the Schrödinger equation is associated to the use of real numbers (or more rigorously to hermiticity) for the evolution of quantum states. It is just when we relax this and consider complex eigenvalues for the Hamiltonian that an irreversible component arises in the evolution. In standard literature, the dissipativity condition is then defined as Im{H} =< 0. Semi-phenomenological models of this class are useful to study the (exponential) decay of instable particles.

What you call the “true heat equation” is only an approximation to more general heat equations that also work for strong gradients of temperature and for systems with memory (non-Markovian).

I want to emphasize that the Schrödinger equation is deterministic. If you know the state of the system at instant t you can know any other instant both in the future or the past. And the classical limit of quantum mechanics is also deterministic and time-reversible.

Past cannot be infinite, because this would imply that time cannot flow. This is explained to broad audiences in the section “How does time evolves?” in the bottom part of the next link

http://www.canonicalscience.org/research/time.html

If anyone needs the technical details, can ask me.

In the quantum world, we can understand unfolding events as a Markov process where time is treated as an absolute parameter. This means that the process is ignorant of the signed value of time. As an example, in a coin flipping scenario, reversing time is not going to produce the past history of coin tosses. IOW, tossing the coin at time -1 will have the same 50-50 probability as tossing the coin at time 1.

What is surprising then is that the Schrodinger equation tells us that it is just as unlikely to reach a particular state in the past as it is to reach a particular state in the future, e.g. the past is just as unpredictable as the future. It is only when we transition into the classical realm, the one mappable to real numbers, that we see time reversal asymmetry and the notion of past and present. There past events have more concrete values, and we have a more solid concept of history.

Thanks.