6 ms·

Is Faster-Than-Light Travel or Communication Possible?

47 points by bhavin 15y ago

asymmetric 15y ago
anyone care to give a simple TL;DR for the rest of us?
- pohl 15y ago
  http://www.desy.de/user/projects/Physics/Relativity/SpeedOfLight/FTL.html#conclusion http://www.desy.de/user/projects/Physics/Relativity/SpeedOfL...
- wheeee 15y ago
  Is FTL travel or communication possible: No.
- nyrath 15y ago
  http://www.projectrho.com/rocket/fasterlight.php#id--Causality http://www.projectrho.com/rocket/fasterlight.php#id--Causali... "Causality, Relativity, FTL travel: chose any two. You cannot have all three." Physicists choose Relativity since it has been tested to many decimal places and always comes through with flying colors. Relativity means that FTL travel is the exact same thing as a time machine. Time machines can cause temporal paradoxes like the Grandfather paradox. Temporal paradoxes render Causality impossible. And without Causality, the entire structure of science crumbles. The only way to have all three is if there is some weird magic law that makes it impossible to use time machines to create temporal paradoxes.
lachenmayer 15y ago
There is one sentence I have a problem with in this post: "Shadows and spotlights suffice to show that there is no logic in this suggestion, because they can certainly go FTL and still be seen." Now, I was only scanning over the article and certainly missed some explanatory section, but can someone explain to me how shadows and spotlights go faster than light?
- carbocation 15y ago
  They don't. These are not truly examples of things traveling from Point A to Point B. No information is being transmitted FTL. It is not clear to me why the author sees this as confusing. Imagine you have a device with two lasers (Laser A and Laser B) that each point down to a white surface. The lasers are one meter from one another, and are calibrated such that Laser B fires exactly 1/(600,000,000)th of a second after Laser A. It might look that the dot is moving faster than light, since light travels at 300,000,000 m/s; however, as we know, nothing went from Point A to Point B.
- civilian 15y ago
  The idea is the angle of it. If I have an intensely bright spotlight (on, say, a satallite?), and I make it spin 360 degrees in a second, then somewhere out in space the trail of light is going faster than the speed of light. You can also think about the beacon that a pulsar gives off. We can detect the pulses happening every few seconds-- so what happens when our friend on Alpha Centauri also detects the pulses? The pulsar 'spotlight' is cycling between Alpha Centauri & Earth in a fraction of a second, even though we're 4 lightyears away from eachother. It's not really FTL, only from a certain point of view. As the author says: "These are all examples of things that can go faster than light, but which are not physical objects. It is not possible to send information faster than light on a shadow or light spot, so FTL communication is not possible in this way. This is not what we mean by faster than light travel, although it shows how difficult it is to define what we really do mean by faster than light travel."
- nadam 15y ago
  Think about the following scenario: you have a point lightsource in the center of a circle. The radius of the circle is R. You have a point on a concentric circle with radius R/2. This point is moving on the circumference of this small circle with the speed of light. If my calculations are right the shadow of the point on the big circle runs with 2*c.
- raganwald 15y ago
  Shadows and spots can appear to go faster than light. Example given in the article: We shine a laser at the moon and wave it gently. The spot of light on the moon moves across the surface faster than light would travel. However, what we call the spot of light on the moon is not an object in any real sense. If I am standing on the Earth with my laser and I place two observers on either edge the moon, we may say colloquially that a spot has moved from one observer to the other faster than light can travel, but in actual fact what has really happened is that light has travelled from me to the first observer and then to the second observer at the speed of light, and it is just that my signal to the second observer arrived extremely soon after my signal to the first observer.
raganwald 15y ago
I am constantly reading that information cannot travel faster than light, and I accept each of the explanations as to why the various methods for attempting to send information faster than light do not work. What I do not understand is whether sending information faster than light would cause a paradox of any sort. It may be that it happens to be impossible. But is it necessarily impossible?
- Confusion 15y ago
  Information can kill (indirectly, for instance by driving someone to commit murder or suicide) and as such it is the grandfather paradox from the article all over again.
- llimllib 15y ago
  This page looks to answer your question with a fairly comprehensive "we're not sure": http://www.desy.de/user/projects/Physics/Relativity/SpeedOfLight/FTL.html#18 http://www.desy.de/user/projects/Physics/Relativity/SpeedOfL... (I can't vouch for its accuracy. It seems that the grandfather paradox is the biggest problem with FTL communication)
  - raganwald 15y ago
    The suggestion seems to be that FTL communication is equivalent to sending information back in time. But the explanation on this page asks me to make massive leaps. I can only imagine it is not meant to explain, but rather as a reminder for people who already understand the explanation.
- martincmartin 15y ago
  But is it necessarily impossible? Yes, because of time dilation. In particular, suppose that, in some reference frame, you can send a message from location A to location B faster than light. In that reference frame, it arrives at B after it leaves A. Then, in some other reference frame (moving at less than the speed of light relative to the first), it arrives at B before it leaves A. So, we can reverse that: Let's say I (in Boston) want to send you (in Toronto) a message that you'll receive before I send it. Then I simply figure out a reference frame where the message would travel forward in time but faster than light. Then I use the faster-than-light communication in that reference frame to send it to you. If you now do the same -- choose a 3rd reference frame, where the message from Toronto to Boston travels forward in time but faster than light, but where on the Earth's surface it travels backward in time -- you can now send your reply before I send my message. Then I can arrange a simple grandmother paradox. e.g. I plan to send a message to you at 2pm, but if I get your reply before 2pm, I don't actually send it. EDIT: In particular, if one observer measures an event as happening at position-time (x, y, z, t), and another observer is moving at speed u along the x axis, that observer will see: x' = (x - ut) / sqrt(1 - u^2/c^2) y' = y z' = z t' = (t - ux/c^2) / sqrt(1 - u^2/c^2) (This is the Lorentz transformation.) Suppose the first observer can send info faster than light. Suppose she sends it along the x axis at a speed of 2c. You can repeat the derivation with any kc where k > 1. If the sending event is at position-time (0, 0, 0, 0), and it takes one time unit to get there, then the receiving event in her frame is (2c, 0, 0, 1). What does the other observer see? Let's suppose they're moving more than half the speed of light, say 3/4 the speed of light. The sending event is the origin, and the origin maps to the origin, so that's easy. For the receiving event: x' = (x - ut) / sqrt(1 - u^2/c^2) = (2c - (3/4 c) * 1) / sqrt(1 - (3/4 c)^2/c^2) y' = y = 0 z' = z = 0 t' = (t - ux/c^2) / sqrt(1 - u^2/c^2) = (1 - (3/4 c) * 2 * c/c^2) / sqrt(1 - (3/4 c)^2 / c^2) = (1 - 3/4 * 2) / sqrt(1 - (3/4)^2). In the formula for t', note that the first term, 1 - 3/4 * 2, is negative. So, if I can send messages at a speed of k * c, then an observer moving with a speed greater than c/k in the same direction sees the message arrive before it leaves. That description is very easy to reverse.
deleted 15y ago
[deleted]
phypi 15y ago
Assuming that space and time collapse into/onto a one dimensional membrane, time wouldn't be of consequence. Information would remain and it would be "every where" at "every time." Question is, where are we in reference to that membrane right now?
- hugh3 15y ago
  Assuming that space and time collapse into/onto a one dimensional membrane And why, exactly, would we assume a thing like that?
VladRussian 15y ago
Special Relativity states (and so far all evidence confirms it ) that speeds faster than "c" is impossible relative to (i.e. _through_ ) local spacetime ("aether") . The mathematical model of SR (smooth manifold with static metric (and we know that our real world doesn't have static metric, so real applications of SR approximate it by considering only sufficiently local regions of space during short periods of time)) doesn't cover and thus doesn't preclude or contradict with other types of relative "motions" (for the lack of better term) which are known to exist and have speed faster than "c" : - quantum entanglement (we don't know the machinery behind it, very-very possible that the SR's smoothness of space is also only a rough approximation of the real world.) Obviously any attempts to find the "signal" between the entangled particles in the conventional sense (something traveling through smooth static spacetime) have so far been and will continue to be futile. - relative motion of the parts of the (our expanding) Universe that are far enough from each other (observed, space expansion, described by General Relativity, at this scales of spacetime the SR's condition of static metric just isn't valid anymore )
geuis 15y ago
I'm going to be the odd man out here. There are several statements made in the article that are incorrect, and then other arguments are based on them. In particular, the tying together of speed and time. "By 'world line' we mean a curve traced out in the four dimensions of space-time". Time is not a dimension. Time is a human concept. Clocks don't measure a physical dimension or force called time, they repeat mechanical motions at regular intervals that we label as time passing. Same for atomic clocks. We measure the spin rates of cesium atoms but that isn't any different than mechanical clocks. Time is not a force that figures into physics equations. Further, time doesn't exist. Not in the classical sense. What we experience as time is the entropy of energy in the universe winding out. We remember things that happened before and imagine things in the future, but since there is no physical "time" dimension there is no traveling forward and back. If you are on a ship traveling close to c, the rate of entropy in the matter and energy on your ship is lower than the outside universe. That's why time seems slower. There's a classic thought experiment that you can build a time machine by putting one end of a wormhole on a ship, send it out for 20 years near c, then bring it back. At this point you would have a wormhole with ends in two different times. It doesn't work like that though. Passing through it will just put you wherever it opens to, and you'll just end up in whatever local entropy rate is going on.
- Chirono 15y ago
  Further, time doesn't exist. Not in the classical sense... If you are on a ship traveling close to c, the rate of entropy in the matter and energy on your ship is lower than the outside universe. That's why time seems slower. Not trying to be facetious, this is a genuine question, but rate of change in entropy with respect to what?
  - felipemnoa 15y ago
    I suspect with respect to the frame of reference from where the measurement is being done. If the frame of reference is the ship itself then you will not notice any change in entropy. Hence why you won't be able to tell that time is actually slower. However, from a frame of reference outside the ship you will notice the difference in rate of change. So really what you are really measuring is the difference in change of entropy from your local frame of reference to the frame of reference of the ship. There is no such thing as absolute rate of change. All is relative. With respect to time, if you could somehow reverse all the motions of every single subatomic particles in a particular frame of reference then you would essentially be moving back in time in that specific frame of reference. To reverse time you have to reverse the motion of every single particle, sub-particle. Is almost the same thing as simply playing a movie in reverse. Now the real issue with this methodology is that the particle movements are not being recorded anywhere as far as we can tell. So we need to first find a way to record the movement of all the particles in a frame of reference and then find another way to run the entire recording in reverse. I wonder if exceeding the speed of light would actually reverse the motion of particles. That would imply that the motion is somehow being recorded? Who knows, just thinking out-loud. Another really good question that I've been wondering about is why does entropy decreases when you move faster? What is it that causes entropy to decrease? Is it some sort of "friction" with space-time? Anybody have any good suggestions? It may have to do something with conservation of energy. Or conservation of something. The faster it moves the slower the particles move. Something is being compensated for.
chicagobob 15y ago
Something I've always wondered is -- of course going back in time is a bad thing -- but is going "faster than light" necessarily going back in time? I wonder why, say, instantaneous communication across the galaxy wouldn't be possible, as long as it doesn't go back in time. (BTW: not necessarily from a physics point of view, they would say that not only can't one travel back in time, neither can you go faster than the speed of light, but I was wondering more from a paradox / time causality point of view).
- hugh3 15y ago
  See my comment elsewhere in this thread, but the answer is yes but I can't explain why in anything shorter than an entire textbook on relativity. If you've "always wondered" about this, then this is the universe telling you to go and learn relativity (at least special relativity).
Groxx 15y ago
From the article: >Now consider the description of an EPR-entangled pair of photons: (|↑↓> + |↓↑>)/√2 At first glance this looks very much like the single-photon case, except that where before we had ΨU and ΨL we now have |↑↓> and |↓↑>, representing respectively photon 1 being in the upper slit and photon 2 being in the lower slit and viceversa. But this distinction is crucial because it turns out that there is some notational sleight-of-hand going on here. First, |↑↓> is shorthand for |↑>|↓>. Second, the arrow symbols have no semantic significance; they are just compact mnemonic identifiers. We could just as well have written |UL> and |LU> (which of course is shorthand for |U>|L> and |L>|U>) as |↑↓> and |↓↑>. Finally, ΨU is just another way of writing |U>. So if we employ alternative notation we get the following description of two entangled photons: (ΨU |U> + ΨL |L>)/√2 As I have probably demonstrated other places, IANAQM. But I don't follow that last transformation. If |↑↓> == |↑>|↓> == |U>|L> == |UL> and ΨU == |U>, how does (|↑↓> + |↓↑>)/√2 == (ΨU |U> + ΨL |L>)/√2 and not (ΨU |L> + ΨL |U>)/√2 ? Shouldn't the |U> and |L> be reversed?

47 points by bhavin 15y ago

asymmetric 15y ago

anyone care to give a simple TL;DR for the rest of us?

pohl 15y ago
http://www.desy.de/user/projects/Physics/Relativity/SpeedOfLight/FTL.html#conclusion http://www.desy.de/user/projects/Physics/Relativity/SpeedOfL...
wheeee 15y ago
Is FTL travel or communication possible: No.
nyrath 15y ago
http://www.projectrho.com/rocket/fasterlight.php#id--Causality http://www.projectrho.com/rocket/fasterlight.php#id--Causali... "Causality, Relativity, FTL travel: chose any two. You cannot have all three." Physicists choose Relativity since it has been tested to many decimal places and always comes through with flying colors. Relativity means that FTL travel is the exact same thing as a time machine. Time machines can cause temporal paradoxes like the Grandfather paradox. Temporal paradoxes render Causality impossible. And without Causality, the entire structure of science crumbles. The only way to have all three is if there is some weird magic law that makes it impossible to use time machines to create temporal paradoxes.

lachenmayer 15y ago

There is one sentence I have a problem with in this post: "Shadows and spotlights suffice to show that there is no logic in this suggestion, because they can certainly go FTL and still be seen." Now, I was only scanning over the article and certainly missed some explanatory section, but can someone explain to me how shadows and spotlights go faster than light?

carbocation 15y ago
They don't. These are not truly examples of things traveling from Point A to Point B. No information is being transmitted FTL. It is not clear to me why the author sees this as confusing. Imagine you have a device with two lasers (Laser A and Laser B) that each point down to a white surface. The lasers are one meter from one another, and are calibrated such that Laser B fires exactly 1/(600,000,000)th of a second after Laser A. It might look that the dot is moving faster than light, since light travels at 300,000,000 m/s; however, as we know, nothing went from Point A to Point B.
civilian 15y ago
The idea is the angle of it. If I have an intensely bright spotlight (on, say, a satallite?), and I make it spin 360 degrees in a second, then somewhere out in space the trail of light is going faster than the speed of light. You can also think about the beacon that a pulsar gives off. We can detect the pulses happening every few seconds-- so what happens when our friend on Alpha Centauri also detects the pulses? The pulsar 'spotlight' is cycling between Alpha Centauri & Earth in a fraction of a second, even though we're 4 lightyears away from eachother. It's not really FTL, only from a certain point of view. As the author says: "These are all examples of things that can go faster than light, but which are not physical objects. It is not possible to send information faster than light on a shadow or light spot, so FTL communication is not possible in this way. This is not what we mean by faster than light travel, although it shows how difficult it is to define what we really do mean by faster than light travel."
nadam 15y ago
Think about the following scenario: you have a point lightsource in the center of a circle. The radius of the circle is R. You have a point on a concentric circle with radius R/2. This point is moving on the circumference of this small circle with the speed of light. If my calculations are right the shadow of the point on the big circle runs with 2*c.
raganwald 15y ago
Shadows and spots can appear to go faster than light. Example given in the article: We shine a laser at the moon and wave it gently. The spot of light on the moon moves across the surface faster than light would travel. However, what we call the spot of light on the moon is not an object in any real sense. If I am standing on the Earth with my laser and I place two observers on either edge the moon, we may say colloquially that a spot has moved from one observer to the other faster than light can travel, but in actual fact what has really happened is that light has travelled from me to the first observer and then to the second observer at the speed of light, and it is just that my signal to the second observer arrived extremely soon after my signal to the first observer.

raganwald 15y ago

I am constantly reading that information cannot travel faster than light, and I accept each of the explanations as to why the various methods for attempting to send information faster than light do not work. What I do not understand is whether sending information faster than light would cause a paradox of any sort. It may be that it happens to be impossible. But is it necessarily impossible?

Confusion 15y ago
Information can kill (indirectly, for instance by driving someone to commit murder or suicide) and as such it is the grandfather paradox from the article all over again.
llimllib 15y ago
This page looks to answer your question with a fairly comprehensive "we're not sure": http://www.desy.de/user/projects/Physics/Relativity/SpeedOfLight/FTL.html#18 http://www.desy.de/user/projects/Physics/Relativity/SpeedOfL... (I can't vouch for its accuracy. It seems that the grandfather paradox is the biggest problem with FTL communication)
- raganwald 15y ago
  The suggestion seems to be that FTL communication is equivalent to sending information back in time. But the explanation on this page asks me to make massive leaps. I can only imagine it is not meant to explain, but rather as a reminder for people who already understand the explanation.
martincmartin 15y ago
But is it necessarily impossible? Yes, because of time dilation. In particular, suppose that, in some reference frame, you can send a message from location A to location B faster than light. In that reference frame, it arrives at B after it leaves A. Then, in some other reference frame (moving at less than the speed of light relative to the first), it arrives at B before it leaves A. So, we can reverse that: Let's say I (in Boston) want to send you (in Toronto) a message that you'll receive before I send it. Then I simply figure out a reference frame where the message would travel forward in time but faster than light. Then I use the faster-than-light communication in that reference frame to send it to you. If you now do the same -- choose a 3rd reference frame, where the message from Toronto to Boston travels forward in time but faster than light, but where on the Earth's surface it travels backward in time -- you can now send your reply before I send my message. Then I can arrange a simple grandmother paradox. e.g. I plan to send a message to you at 2pm, but if I get your reply before 2pm, I don't actually send it. EDIT: In particular, if one observer measures an event as happening at position-time (x, y, z, t), and another observer is moving at speed u along the x axis, that observer will see: x' = (x - ut) / sqrt(1 - u^2/c^2) y' = y z' = z t' = (t - ux/c^2) / sqrt(1 - u^2/c^2) (This is the Lorentz transformation.) Suppose the first observer can send info faster than light. Suppose she sends it along the x axis at a speed of 2c. You can repeat the derivation with any kc where k > 1. If the sending event is at position-time (0, 0, 0, 0), and it takes one time unit to get there, then the receiving event in her frame is (2c, 0, 0, 1). What does the other observer see? Let's suppose they're moving more than half the speed of light, say 3/4 the speed of light. The sending event is the origin, and the origin maps to the origin, so that's easy. For the receiving event: x' = (x - ut) / sqrt(1 - u^2/c^2) = (2c - (3/4 c) * 1) / sqrt(1 - (3/4 c)^2/c^2) y' = y = 0 z' = z = 0 t' = (t - ux/c^2) / sqrt(1 - u^2/c^2) = (1 - (3/4 c) * 2 * c/c^2) / sqrt(1 - (3/4 c)^2 / c^2) = (1 - 3/4 * 2) / sqrt(1 - (3/4)^2). In the formula for t', note that the first term, 1 - 3/4 * 2, is negative. So, if I can send messages at a speed of k * c, then an observer moving with a speed greater than c/k in the same direction sees the message arrive before it leaves. That description is very easy to reverse.

deleted 15y ago

[deleted]

phypi 15y ago

Assuming that space and time collapse into/onto a one dimensional membrane, time wouldn't be of consequence. Information would remain and it would be "every where" at "every time." Question is, where are we in reference to that membrane right now?

hugh3 15y ago
Assuming that space and time collapse into/onto a one dimensional membrane And why, exactly, would we assume a thing like that?

VladRussian 15y ago

Special Relativity states (and so far all evidence confirms it ) that speeds faster than "c" is impossible relative to (i.e. _through_ ) local spacetime ("aether") . The mathematical model of SR (smooth manifold with static metric (and we know that our real world doesn't have static metric, so real applications of SR approximate it by considering only sufficiently local regions of space during short periods of time)) doesn't cover and thus doesn't preclude or contradict with other types of relative "motions" (for the lack of better term) which are known to exist and have speed faster than "c" : - quantum entanglement (we don't know the machinery behind it, very-very possible that the SR's smoothness of space is also only a rough approximation of the real world.) Obviously any attempts to find the "signal" between the entangled particles in the conventional sense (something traveling through smooth static spacetime) have so far been and will continue to be futile. - relative motion of the parts of the (our expanding) Universe that are far enough from each other (observed, space expansion, described by General Relativity, at this scales of spacetime the SR's condition of static metric just isn't valid anymore )

geuis 15y ago

I'm going to be the odd man out here. There are several statements made in the article that are incorrect, and then other arguments are based on them. In particular, the tying together of speed and time. "By 'world line' we mean a curve traced out in the four dimensions of space-time". Time is not a dimension. Time is a human concept. Clocks don't measure a physical dimension or force called time, they repeat mechanical motions at regular intervals that we label as time passing. Same for atomic clocks. We measure the spin rates of cesium atoms but that isn't any different than mechanical clocks. Time is not a force that figures into physics equations. Further, time doesn't exist. Not in the classical sense. What we experience as time is the entropy of energy in the universe winding out. We remember things that happened before and imagine things in the future, but since there is no physical "time" dimension there is no traveling forward and back. If you are on a ship traveling close to c, the rate of entropy in the matter and energy on your ship is lower than the outside universe. That's why time seems slower. There's a classic thought experiment that you can build a time machine by putting one end of a wormhole on a ship, send it out for 20 years near c, then bring it back. At this point you would have a wormhole with ends in two different times. It doesn't work like that though. Passing through it will just put you wherever it opens to, and you'll just end up in whatever local entropy rate is going on.

Chirono 15y ago
Further, time doesn't exist. Not in the classical sense... If you are on a ship traveling close to c, the rate of entropy in the matter and energy on your ship is lower than the outside universe. That's why time seems slower. Not trying to be facetious, this is a genuine question, but rate of change in entropy with respect to what?
- felipemnoa 15y ago
  I suspect with respect to the frame of reference from where the measurement is being done. If the frame of reference is the ship itself then you will not notice any change in entropy. Hence why you won't be able to tell that time is actually slower. However, from a frame of reference outside the ship you will notice the difference in rate of change. So really what you are really measuring is the difference in change of entropy from your local frame of reference to the frame of reference of the ship. There is no such thing as absolute rate of change. All is relative. With respect to time, if you could somehow reverse all the motions of every single subatomic particles in a particular frame of reference then you would essentially be moving back in time in that specific frame of reference. To reverse time you have to reverse the motion of every single particle, sub-particle. Is almost the same thing as simply playing a movie in reverse. Now the real issue with this methodology is that the particle movements are not being recorded anywhere as far as we can tell. So we need to first find a way to record the movement of all the particles in a frame of reference and then find another way to run the entire recording in reverse. I wonder if exceeding the speed of light would actually reverse the motion of particles. That would imply that the motion is somehow being recorded? Who knows, just thinking out-loud. Another really good question that I've been wondering about is why does entropy decreases when you move faster? What is it that causes entropy to decrease? Is it some sort of "friction" with space-time? Anybody have any good suggestions? It may have to do something with conservation of energy. Or conservation of something. The faster it moves the slower the particles move. Something is being compensated for.

chicagobob 15y ago

Something I've always wondered is -- of course going back in time is a bad thing -- but is going "faster than light" necessarily going back in time? I wonder why, say, instantaneous communication across the galaxy wouldn't be possible, as long as it doesn't go back in time. (BTW: not necessarily from a physics point of view, they would say that not only can't one travel back in time, neither can you go faster than the speed of light, but I was wondering more from a paradox / time causality point of view).

hugh3 15y ago
See my comment elsewhere in this thread, but the answer is yes but I can't explain why in anything shorter than an entire textbook on relativity. If you've "always wondered" about this, then this is the universe telling you to go and learn relativity (at least special relativity).

Groxx 15y ago

From the article: >Now consider the description of an EPR-entangled pair of photons: (|↑↓> + |↓↑>)/√2 At first glance this looks very much like the single-photon case, except that where before we had ΨU and ΨL we now have |↑↓> and |↓↑>, representing respectively photon 1 being in the upper slit and photon 2 being in the lower slit and viceversa. But this distinction is crucial because it turns out that there is some notational sleight-of-hand going on here. First, |↑↓> is shorthand for |↑>|↓>. Second, the arrow symbols have no semantic significance; they are just compact mnemonic identifiers. We could just as well have written |UL> and |LU> (which of course is shorthand for |U>|L> and |L>|U>) as |↑↓> and |↓↑>. Finally, ΨU is just another way of writing |U>. So if we employ alternative notation we get the following description of two entangled photons: (ΨU |U> + ΨL |L>)/√2 As I have probably demonstrated other places, IANAQM. But I don't follow that last transformation. If |↑↓> == |↑>|↓> == |U>|L> == |UL> and ΨU == |U>, how does (|↑↓> + |↓↑>)/√2 == (ΨU |U> + ΨL |L>)/√2 and not (ΨU |L> + ΨL |U>)/√2 ? Shouldn't the |U> and |L> be reversed?