Last time I opened with basic exponentiation and raised it to the idea of complex exponents (which may, or may not, have been surprising to you). I also began exploring the ubiquitous exp function, which enables the complex math needed to deal with such exponents.
The exp(x) function, which is the same as ex, appears widely throughout physics. The complex version, exp(ix), is especially common in wave-based physics (such as optics, sound, and quantum mechanics). It’s instrumental in the Fourier transform.
Which in turn is as instrumental to mathematicians and physicists as a hammer is to carpenters and pianos.
Sunday I breezed through Seven Brief Lessons On Physics (2014), by Carlo Rovelli. It’s a quick read of only 96 pages that still manages to touch on some of the key aspects of physics.
His much longer book, Reality Is Not What It Seems: The Journey to Quantum Gravity (2014), covers the same territory in greater detail (and greater length: 288 pages). After I finished what amounted to an appetizer, I tucked into the main course. I’m about 30% through it and am enjoying it quite a bit more than I have his work so far.
Both books, but especially the longer one, explore the theory of Loop Quantum Gravity (LQG), of which Rovelli is a co-founder.
I’ve posted more than once regarding my view of the Many Worlds Interpretation (MWI) of quantum physics. I find its rise in modern popularity genuinely inexplicable. (I can’t help but think it’s exactly the sort of thing Dr. Sabine Hossenfelder is talking about in her book, Lost in Math.)
Hoping to find the logic that apparently appeals to so many, I read Something Deeply Hidden: Quantum Worlds and the Emergence of Spacetime (2019), by Sean Carroll. It is, in large part, his argument favoring the MWI. Carroll is a leading voice in promoting the view, so I figured his book would address my concerns.
But as far as I can tell, “there is no there there.”
Back in 2015, to celebrate Albert Einstein’s birthday, I wrote a month-long series of posts about Special Relativity. I still regard it as one of my better efforts here. The series oriented on explaining to novices why faster-than-light travel (FTL) is not possible (short answer: it breaks reality).
So no warp drive. No wormholes or ansibles, either, because any FTL communication opens a path to the past. When I wrote the series, I speculated an ansible might work within an inertial frame. A smarter person set me straight; nope, it breaks reality. (See: Sorry, No FTL Radio)
Then Dr Sabine Hossenfelder seemed to suggest it was possible.
Last time I started talking about entropy and a puzzle it presents in cosmology. To understand the puzzle we have to understand entropy, which is a crucial part of our view of physics. In fact, we consider entropy to be a (statistical) law about the behavior of reality. That law says: Entropy always increases.
There are some nuances to this, though. For example we can decrease entropy in a system by expending energy. But expending that energy increases the entropy in some other system. Overall, entropy does always increase.
This time we’ll see how Roger Penrose, in his 2010 book Cycles of Time, addresses the puzzle entropy creates in cosmology.
I’ve been chiseling away at Cycles of Time (2010), by Roger Penrose. I say “chiseling away,” because Penrose’s books are dense and not for the fainthearted. It took me three years to fully absorb his The Emperor’s New Mind (1986). Penrose isn’t afraid to throw tensors or Weyl curvatures at readers.
This is a library book, so I’m a little time constrained. I won’t get into Penrose’s main thesis, something he calls conformal cyclic cosmology (CCC). As the name suggests, it’s a theory about a repeating universe.
What caught my attention was his exploration of entropy and the perception our universe must have started with extremely low entropy.
Last time I started with wave-functions of quantum systems and the Schrödinger equation that describes them. The wave-like nature of quantum systems allows them to be merged (superposed) into combined quantum system so long as the coherence (the phase information) remains intact.
The big mystery of quantum wave-functions involves their apparent “collapse” when an interaction with (a “measurement” by) another system seemingly destroys their coherence and, thus, any superposed states. When this happens, the quantum behavior of the system is lost.
This time I’d like to explore what I think might be going on here.
Quantum physics is weird. How weird? “Too weird for words,” as we used to say, and there is a literal truth to words being inadequate in this case. There is no way to look at the quantum world that doesn’t break one’s mind a little. No one truly understands it (other than through the math). It’s like trying to see inside your own head.
Since we’re clueless we make up stories to fit the facts. Some stories advise that we just keep our heads down and do the math. (Which works very well but leaves us thirsty.) Other stories seek to quench that thirst, but every story seems to stumble somewhere.
One of quantum’s biggest and oldest stumbling blocks is wave-function collapse.
I’ve come to realize that, when it comes to the Many Worlds Interpretation (MWI) of quantum physics, there is at least one aspect of it that’s poorly understood. Since it’s an aspect that even proponents of MWI recognize as an issue, I thought I’d take a stab at explaining it. (If nothing else, I’ll have a long reply I can link to in the future.)
The issue in question involves what MWI does to probability. Essentially, our view of rare events — improbable events — is that they happen rarely, as we’d expect. Flip a fair coin 100 times; we expect to get heads roughly 50% of the time.
But under MWI, someone always gets 100 heads in a row.
Last week, when I posted about the Mathematical Universe Hypothesis (MUH), I noted that it has the same problem as the Block Universe Hypothesis (BUH): It needs to account for its apparent out-of-the-box complexity. In his book, Tegmark raises the issue, but doesn’t put it to bed.
He invokes the notion of Kolmogorov complexity, which, in a very general sense, is like comparing things based on the size of their ZIP file. It’s essentially a measure of the size of information content. Unfortunately, his examples raised my eyebrows a little.
Today I thought I’d explore why. (Turns out I’m glad I did.)