Kind: captions
Language: en
what is hilbert space and euclidean
space yeah you know I think that people
are very welcome to go through their
lives not knowing what Hilbert spaces
but if you if you what I dig in a little
bit more into quantum mechanics it
becomes necessary you know the English
language was invented long before
quantum mechanics or various forms of
higher mathematics were invented so we
use the word space to mean different
things of course most of us think of
space as this three dimensional world in
which we live right I mean some of us
just think of it as outer space okay but
space around us it gives us the
three-dimensional location of things and
objects but mathematicians use any
generic abstract collection of elements
as a space okay a space of possibilities
you know momentum space etc so Hilbert
space is the space of all possible
quantum wave functions either for the
universe or for some specific system and
it could be an infinite dimensional
space or it could be just really really
large dimensional but finite we don't
know because we don't know the final
theory of everything but this abstract
Hilbert space is really really really
big and has no immediate connection to
the three-dimensional space in which we
live what what do dimensions in Hilbert
space mean you know it's just a way of
mathematically representing how much
information is contained in the state of
the system how many numbers do you have
to give me to specify what the thing is
doing so in classical mechanics I give
you the location of something by giving
you three numbers right up down like XYZ
coordinates but then I might want to
give you its entire state physical state
which means both its position and also
its velocity the velocity also has three
components so it's state lives in
something called phase space which is
six dimensional three dimensions of
position three dimensions of velocity
and then if it also has an orientation
in space that's another three dimensions
and so forth so as you describe more and
more information about the system you
have an abstract mathematical space that
has more and more numbers that you need
to give and each one of those numbers
corresponds to a dimension in that space
so in terms of not amount of information
what is entropy this mystical word
that's overused in math and physics but
has a very specific meaning in this
context sadly it has more than one very
specific meeting this is this is reason
why it is hard and Rumi means different
things even to different physicists but
one way of thinking about it is a
measure of how much we don't know about
the state of a system right so if I have
a bottle of water molecules and I know
that okay there's a certain number of
water molecules I could weigh it right
and figure out I know the volume of it
and I know the temperature and pressure
and things like that I certainly don't
know the exact position and velocity of
every water molecule right so there's a
certain amount of information I know
certain amount that I don't know that is
that is part of the complete state of
the system and that's what the entropy
characterizes how much unknown
information there is the difference
between what I do know about the system
and is full exact microscopic state so
when we try to describe a quantum
mechanical system is it infinite or
finite but very large yeah we don't know
that depends on the system you know it's
easy to mathematically write down a
system that would have a potentially
infinite entropy an infinite dimensional
hilbert space so let's let's go back a
little bit we said that the hilbert
space was the space in which quantum
wave functions lived for different
systems that will be different sizes
they could be infinite or finite so
that's the number of numbers the number
of pieces information you could
potentially give me about the system so
the bigger Hilbert spaces the bigger the
entropy of that system could be
depending what I know about it if I
don't know anything about it then you
know has a huge entropy right but only
up to the size of its Hilbert space so
we don't know in in the real physical
world whether or not you know this
region of space that contains that water
bottle has potentially an infinite
entropy or just a finite entropy we have
we have different arguments on different
sides so if it's infinite how do you
think about infinity is this something
you can your cognitive abilities are
able to process or is it just a
mathematical tool it's somewhere in
between right I mean we can say things
about it we can use mathematical tools
to manipulate infinity very very
accurately you can define what we mean
you know for any number n there's a
number bigger than it so there's no
biggest number right so there's
something called the total number of all
numbers that's infinite but it is hard
to wrap your brain around that and I
think that gives people pause because we
talk about infinity as if it's a number
but it has plenty of properties that
real numbers don't have you know if you
multiply infinity by two you get
infinity again right that's a little bit
different than what we're used to okay
but are you comfortable with the idea
that in thinking of what the real world
actually is that infinity could be part
of that world
are you comfortable that a world in some
dimension and somehow comfortable with
lots of things I mean you know I I don't
want my level of comfort to affect what
I think about the world you know I'm
pretty open-minded about what the world
could be at the fundamental level yeah
but infinity is a is a tricky one it's
not almost a question of comfort it's a
question of is it an overreach of our
intuition sort of it could be a
convenient almost like when you add a
constant to an equation just because
it'll help it just feels like it's
useful to at least be able to imagine a
concept not directly but in some kind of
way that this feels like it's a
description of the real world think of
it this way there's only three numbers
that are simple there's zero there's one
and there's infinity a number like 318
it's just bizarre like that that you
need a lot of bits to give me what that
number is
yeah but zero and one infinity like once
you have 300 things you might as well
have infinity things right otherwise you
have to say how when to stop making the
thing that's right so there's a sense in
which infinity is a very natural number
of things to exist that I was never
comfortable with because it's just such
a good but it was a too good to be true
mmm because in math it just helps make
things work out when things get very
it's when things get very large close
infinity things seem to work out nicely
it's kind of like because of my deepest
passion it's probably psychology and I'm
uncomfortable how in the average the the
beauty of the very very the how much we
very is lost in that same kind of sense
infinity seems like convenient way to
erase the details but the thing about
infinity is it seems to pop up whether
we like it or not right right like
you're trying to be a computer scientist
you ask yourself well how long will it
take this program to run and you realize
well for some of them the answer is
infinitely long it's not because you
tried to get there you wrote a five line
computer program it doesn't halt
you