The h Bar

enumaris

12:00 AM

if RT=0 implies either R=0 or T=0 right...I see no reason for that condition given only R+T=1

Mohammad Areeb Siddiqui

how can the probability of photon being reflected and transmitted at the same time be not 0??

P(reflec) + P(transmit) = 1

P(reflect and transmit) = P(reflect)P(transmit)

= RT

shouldnt it be = 0?

they both are independent events

Avnish Kabaj

2:11 AM

ioc.ee/~kalda/ipho

Are Jaan Kalda's ipho study materials one of a kind

Because they are absolutely brilliant

But sadly not all topics are covered

Secret

2:25 AM

[Random]

What does it mean for something to have no future:

Consider an observer O with some light come J. Let J+ be the future half of the light come

Normally, general relativity requires all dynamics to be deterministic, thus any future event p of O is already determined by the past light cone J-(p)

((Except for very pathological cases as detailed in 0celo's blog Einstein and The Evidence

)

But consider the topology of spacetime has a structure such that no event in $J^+(O)$ is connected to their respective future light cones $\{J^+\}$, then we have these events not the cause of anything in the future including $J^+(O)$ and thus they effectively disconnected with their future

For any such event $q$ such that for any event $p$ in $J^+(O)$ where $|p-q| < \epsilon$, $J^-(p) = \varnothing$

Likewise for $q$, $J^+(q) = \varnothing$

$q$ is thus an example of an event that has no future and the ps that mentioned earlier are examples of events that has no past

These are the basic units you need in order to construct a spacetime such that there are non deterministic regions of dynamics, without the spacetime being eternally expanding or inside a black hole to generate a Cauchy horizon

It is unclear what will be observed by any observer locating at $q$, will they be erased from existence some infinitesimal proper time later, or something else

I think I would love to learn more about coordinate free versions of GR, this is so convenient

Semiclassical

3:27 AM

@MohammadAreebSiddiqui the events "the incident photon is reflected" and "the incident photon is not reflected" are certainly not independent.

It's really not different than the situation for a coin flip. If P(H)=P(T)=1/2, then P(H)P(T)=1/4 is not zero. But H and T are not independent outcomes, so P(H and T) != P(H)P(T)

(Or, even more simply: Mutually exclusive events are not independent.)

enumaris

3:48 AM

@Secret I think there's an issue of is such a causal structure allowed

Secret

@enumaris Well, I don't see how it cannot be done by making the point p in question to be topologically disconnected from it's future or past light cone by e.g. removing a small annular region near p

enumaris

Well, a manifold in GR is a differentiable one...it has to be locally Minkowski. If you start removing regions from the manifold willy nilly, I feel like you will create boundaries along which the "manifold" is no longer a manifold in that sense

I have no proof of this, but I think some care should be taken

Secret

Ah right, yes, once sections start to be removed, it will ceased to be a manifold. hmm...

John Rennie

I don't think there's any problem removing bits from a manifold. It's just that you will now have incomplete geodesics.

enumaris

yeah, from a physical perspective, it would be like you couldn't look left all of a sudden cus you ran up to a boundary or something...how odd would that be...lol

well I think there could be ways in which you can remove bits from a manifold...but the way in which they are removed probably has to be special

John Rennie

4:03 AM

A black hole is an example where geodesics are future incomplete and a white hole is an example where geodesics are past incomplete.

enumaris

well they are incomplete because of the presence of the singularity

the singularity itself is not generally thought of as part of the manifold

John Rennie

Yes, so we have a manifold with a bit missing.

enumaris

a singularity missing...and the singularity itself is...theoretically...difficult

Sir Cumference

@JohnRennie I have to ask, how'd you find that?

enumaris

One should not remove parts of a manifold without care is what I'm getting at. You would have to prove that removing these pieces makes mathematical and physical sense.

John Rennie

4:05 AM

@SirCumference someone else mentioned it :-)

enumaris

there is great care in dealing with the singularities present in physics. And a lot of work has gone into trying to hide singularities behind event horizons...

if you just remove a chunk of the manifold...that's like...pretty catastrophic imo T_T

Secret

tbh, I really have no idea what happens when something enters a nondeterministic region, because any of the following can happen:

1. The object gets erased from existence because it "fell out of spacetime"

2. The object's present state cannot be inferred from its past states once a threshold is crossed (that's the case for Cauchy horizons)

3. The object disappears and then reappeared elsewhere

enumaris

I think when you start to get to issues as esoteric as like "falling out of spacetime", you should start to evaluate whether your theory still has predictive power in that regime.

I wouldn't trust a mathematical result like that without some physical evidence.

It's not to say such weird events can not occur

Secret

news.berkeley.edu/2018/02/20/some-black-holes-erase-your-past

This is why I often found these results strange

John Rennie

Just because an object's future trajectory isn't integrable doesn't mean we need resort to the supernatural :-)

enumaris

4:17 AM

Yeah I saw that news a few weeks ago

I'm not well versed in GR enough to comment either way

at this point it seems to be more of a math question imo

Secret

Sometime later I need to revise just how many nondeterministic scenarios of spacetime can occur because they are quite interesting

Although likely to not be physically relevant to us alot

enumaris

well there are "spacetimes" that have closed timelike curves

like the Godel spacetime

it doesn't surprise me that some other caustics can occur in GR

if global hyperbolicity has to be taken as a postulate of GR, I'm fine with that :P

the question would then be reduced to physically finding a situation where global hyperbolicity breaks down

I'm...not actually sure if the presence of singularities breaks down global hyperbolicity

intuitively it would seem to be yes but I'm not sure

frogeyedpeas

5:42 AM

howdy, i got some elementary questions on bra-ket notation, but reading the wiki article didn't make it obvious so I was wondering if i could ask here

Does $\langle a | b | c \rangle = \int a \circ b \circ c $ ?

John Rennie

@frogeyedpeas What does the $\circ$ operator mean here?

Secret

You don't have the integral sign unless you express the (co)vectors a,c and linear operator b in some continuous basis

enumaris

one of the issues with learning Bra-ket notation from a source like Griffiths that starts from a "physical" perspective is that one is often stuck in viewing things in terms of the position (or momentum) representation. A book like Ballentine that approaches things from much more of a mathematical perspective will be better for that issue. However...there is something to be said about keeping an eye on the physical implications of a theory and not getting too bogged down in the math.

Albas

I like dirac's book the best for bra-ket(after all he created it)

Except the lack of problems, that book is the most clear

Secret

yeah, bra-kets are equally fine in finite dimensional spaces, which is why Susskind introduce them in the context of qubits

enumaris

5:53 AM

interestingly I have not read Dirac's treatment lol

there are a great many books that I should read that I have not...

but right now I have to finish this book on Reinforcement Learning as that has a direct effect on my livelihood

(sorta)

John Rennie

@frogeyedpeas hello?

enumaris

heh, this convo is a bit...4 sided...

needs to be 5 sided though

Secret

Why stop at 4? If I were you I will go full on $\mathfrak{c}$ :P

::finiteness is soooooooo overrated::

enumaris

well there were 4 users who responded lol

Anonymous

6:22 AM

@enumaris Name of the book?

enumaris

which one?

Ballentine?

amazon.com/Quantum-Mechanics-Modern-Development-2nd/dp/…

Anonymous

@enumaris The RL which supposedly has a direct influence on your livelihood. :)

enumaris

oh

Barto and Sutton Reinforcement Learning

I'm reading a pre-print that they released

for edition 2

edition 1 is pretty old

Anonymous

Thanks. I don't have much experience in RL. But have been studying some deep learning stuff in the past few weeks. What's the basic concept in reinforcement learning though? A reward and punishment system...sort of (in order to reduce the error)?

Anonymous

Gotta pick up RL sometime

enumaris

6:26 AM

it's almost exclusively formulated in terms of a markov decision process

state -> action -> reward (or punishment) -> new state

there's an environment which an agent interacts with. The agent has a policy that selects an action given a state, the feedback form the environment is the reward and the next state

Anonymous

But how do you decide the reward/punishment? Are you given the ideal case ?

enumaris

so you have an string of S-A-R-S-A-R-S-...

oh, the reward and punishment should be set by the environment

e.g. if you play space invaders the reward would be your points

it's not set by the learning agent

that would defeat the purpose

but there is some reward engineering you can do depending on your task

Anonymous

Okay. Interesting, I see. So it's not a gradient descent type of thing.

enumaris

You can use gradient descent methods with it

that's if you use function approximation

or if you use policy gradients

Anonymous

I see. Gotta read :)

Anonymous

6:30 AM

Thanks for the overview

enumaris

yeah np :D

you can integrate deep learning into RL

then you get deep-RL

which is what Alphago did to beat the world champ

Anonymous

Haha. Sounds cool

enumaris

Basically the function approximation part you use a deep neural net

so you have an intermediate stage S->f(S)->A where f(S) is some function approximation of S so you don't have to worry about learning a separate value for every state S

f(S) allows you to generalize results from one S to many other states

so S = image of arcade game, f(S) = convnet(image)

instead of say S=every pixel and then choosing an action directly based on the information from each individual pixel

bam, you got the 30 second overview of Deep-RL. There's a ton of other techniques you gotta use to actually make it effective though...XD

frogeyedpeas

@jo

@JohnRennie its operator composition, $a,b,c$ are functions of functions of the domain of the integral. Example $ a = \psi(x) \times \text{arg}$ $b = x \times \text{arg}$ and $ c = \psi^*(x)$ then by my definition $\lange a | b | c \rangle = \int \psi(x) x \psi^*(x) $

enumaris

That...makes no sense... D:

frogeyedpeas

6:47 AM

I don't like my syntax actually now that I think about it, i'm trying to be very abstract and expressive but its more probably more confusing than it needs to be

John Rennie

In $\langle a | \hat{b} | c \rangle$ a and c are wavefunctions and $\hat{b}$ is a hermitian operator, so I'm not sure what you mean by $\psi(x) \times arg$ means.

frogeyedpeas

@JohnRennie what would be the literal definition of that expression in terms of sums or integrals?

I would like to start from a clean slate

John Rennie

$\langle a | \hat{b} | c \rangle = \int a \hat{b}(c) $

frogeyedpeas

okay cool, so thats what i thought but wasn't totally sure. Thank you! :)

John Rennie

I've used $\hat{b}(c)$ to make it clear the operator $\hat{b}$ is acting on the function $c$. Normally we'd just write $\int a \hat{b} c$

enumaris

6:49 AM

if the position-representation of $|a\rangle=\psi(x)$ and the position representation of $|c\rangle=\phi(x)$ then $\langle c|b|a\rangle = \int \phi^*(x)b_x\psi(x)dx$ where $b_x$ is the position representation of $b$

frogeyedpeas

hmm so @enumaris I noticed your definition and @JohnRennie differ since you have complex conjugation in your integral, which he doesn't

enumaris

I should

John Rennie

Oops, yes, my mistake. It should be $\int a^* \hat{b} c$

enumaris

I missed the bra ket on the a and c sorry

see my updated post for more clarity

frogeyedpeas

ah now i see, cool and good :)

enumaris

6:52 AM

but note that the position representation is just one of many possible representations

so it's not always to your benefit to think of the object $\langle c|b|a\rangle$ in terms of a specific representation

frogeyedpeas

Is the rule of complex conjugation specific to a representation?

enumaris

well...hmm...how best to answer that...

I would say no. The complex conjugation part arises when you turn a ket into a bra

the duality between a state $a|a\rangle$ for some complex number $a$ is $a^*\langle a| \leftrightarrow a|a\rangle$

I feel like...the more I explain...the more there will be to explain...

lol

It might be best to refer to some lecture notes or refer to a textbook

e.g. Ballentine

Anonymous

@frogeyedpeas The corresponding member of a ket vector in the dual space is the bra vector which turns out to be the Hermitian conjugate of the ket. The interesting fact is that each vector of a Hilbert space has a one-to-one correspondence to members of the dual space (which is unique).

enumaris

yes...

but now you have to explain what a dual space is...and what a Hilbert space is...

T_T

it never ends...

Anonymous

You always have Wikipedia :P

enumaris

7:00 AM

I find wikipedia to be really good as a reference or as a refresher if you've already learned a subject before...but very rarely is it a good first source

the explanations tend to be too technical

Anonymous

Wikipedia + Stack Exchange is good combination :)

Anonymous

If you're stuck, just ask

Anonymous

Or else, of course there are textbooks

Anonymous

The best thing about Wikipedia is that it does not dilute the details (most of the times)

enumaris

yeah I suppose. I find that in my experience, for all these issues of Hilbert spaces and the like Ballentine was the best source.

Anonymous

7:03 AM

There are over a 1000 introductory QM books I guess :P

frogeyedpeas

I am digesting this all slowly, I think I will get a copy of ballentine though if it is highly recommended

enumaris

the trade-off though is that Ballentine, at least for the first several chapters, reads like a math text rather than a physics one

It doesn't really get to a physical problem for a long while...it spends a lot of time developing the mathematical formalism of QM

frogeyedpeas

I would love something like math text devoid of physical meaning, and then a follow up book that is much more about intuition.

enumaris

well then, Ballentine might be the one for you :D

frogeyedpeas

actually the dream would be to just learn QFT as a mathematical framework directly

enumaris

7:04 AM

QFT is a different beast...

Anonymous

My favourite was Shankar. But if you're an absolute beginner I very highly recommend not learning QM directly before classical mechanics. The other alternative might be to go down the quantum computing path (which is finite-dimensional) and much easier to grasp.

enumaris

Ballentine is normal QM

QFT can suck it

frogeyedpeas

I've been down the quantum computing path, my particular interest is in a (currently ill defined) question that inviolves heisenbergs uncertainty principle and relativity

enumaris

along with every single QFT textbook I've ever read

frogeyedpeas

*i've been down very briefly

*i can only factor some shit, and explain some basic quantum algos and circuits, i think i spoke a bit more pompously than i should have

Slereah

7:06 AM

morning

@frogeyedpeas just read PCT then :p

Anonymous

What's PCT?

frogeyedpeas

@Slereah "https://mathoverflow.net/questions/27701/proof-of-pct-theorem-for-haag-kastler-‌nets-in-qft">

?

i'm not confident if that is a book, a theorem, an article or something else, do you have a link @Slereah?

Anonymous

16

Q: How to learn QFT from mathematical perspective?

I want to learn QFT, because I have heard of its applications in mathematics, I am not interested in scattering cross sections and such. Where can I start to learn? Only books I found are either way above my level or too physics oriented. I know undergraduate math level and have had courses in QM...

Transcript for