5 ms·

Thanks for the explanation. If you have the time, can you also explain _why_ would we be multiplying "each point of the universe with a different complex number

Thanks for the explanation. If you have the time, can you also explain _why_ would we be multiplying "each point of the universe with a different complex number of modulo 1?" What does it mean in physical reality; why multiply points with any number at all?

ufo_pilot 5y ago
Not GP, or a physicist, but my understanding is that the different number at each point you multiply with represents a degree of freedom at each point of spacetime, and in this degree of freedom is where the electromagnetic field lives.
tagrun 5y ago
To find out what kind of symmetry a field (as described by a Lagrangian) admits. It's called gauge symmetry: https://en.wikipedia.org/wiki/Gauge_theory https://en.wikipedia.org/wiki/Gauge_theory It's a complex number for QED but in general, it's a unitary matrix: a rotation which preserves the magnitude of a wavefunction (a complex vector). In physics, symmetries play a fundamental role.
cshimmin 5y ago
The simple answer is "because we can". In general, physicists have found that we should write down the most general mathematical theory compatible with what's observed. A famous example of this is Einstein's cosmological constant -- I'll leave that one to wikipedia [1] since I'm a particle physicist and not an expert on GR. In the case of gauge theory, the idea that we should consider the most general case has been well proven. As GP pointed out, all observable phenomena ultimately depend only on the absolute modulus of the field Ψ, so a theoretical physicist naturally wonders, what happens if you allow its complex phase to vary. Turns out nothing interesting happens if you apply a global phase, but if you allow the phase to vary at every point in spacetime, it ends up breaking the theory. That is, unless you include an additional field at every point in spacetime that precisely cancels out the change induced by the gauge freedom. In other words, the motivation is that we can't simply look and "see" whether or not there is a locally varying phase on the wavefunction Ψ, since we only can measure |Ψ|^2. So we have to assume there is, until proven otherwise. Since a local phase would imply the existence of an extra field to cancel it out, we can indirectly check for this scenario by looking for the corresponding field. As pointed out by GP, in the case of a U(1) gauge, it turns out there is such a field, and electromagnetism (and all of its laws) exactly fit the bill. There are other "unmeasurable" symmetries you could apply to the wave function as well, beyond just a complex phase. SU(2) is a Lie group symmetry which would mean that the measurable properties of certain tuples of fields (Ψ, ϕ) are indistinguishable under a sort of complex-valued rotation of Ψ->ϕ and ϕ->Ψ. Again, if you assume a such symmetry is locally varied at every point in spacetime, you end up requiring not one but three new fields to cancel out the effects on the SM Lagrangian. It turns out that the vector bosons W+, W-, and Z, which mediate weak nuclear forces exactly fit the bill. [1] - https://en.wikipedia.org/wiki/Cosmological_constant https://en.wikipedia.org/wiki/Cosmological_constant
- cominous 5y ago
  Wow you blew my mind. You need to consider creating YouTube videos about this.
- cshimmin 5y ago
  And at the risk of muddying the original point, it turns out that strictly speaking the neither the neutral Z boson nor the electromagnetic photon can be said to come from SU(2) or U(1). Instead, each field is a linear combination of some abstract fields (namely the neutral W0 and the weak hypercharge B bosons) from the unitary gauge induced by the compound symmetry SU(2)xU(1). This is because when both symmetries SU(2) and U(1) are present, there are different ways you can "mix" the two into the Lagrangian. The mixing that gives us the photon and the Z0 boson is known because of the experimental confirmation of these particles. This is what is meant when it is said that electromagnetism and weak neuclear forces are united in a higher-energy theory as a single "electroweak" force.
  - trenchgun 5y ago
    Thank you!
  - jasonwatkinspdx 5y ago
    Thanks for these comments. This simplified a whole bunch of things I had questions about.
  - tinco 5y ago
    This is great, you perfectly bridged the gap between having been 25 once and vaguely remembering the math terms mentioned and bringing it back to reality. Really epic how this (at least to me) abstract and arcane math not only explains the behaviors of the subatomic particles, but even predicts the existence of more and their related forces. The work of theoretical physicists always seemed like the other side of an ocean of math away, but thanks to this explanation it feels like I can at least make out a lighthouse in the distance.
- vehementi 5y ago
  > That is, unless you include an additional field at every point in spacetime that precisely cancels out the change induced by the gauge freedom. Could you elaborate on this a bit? To a layperson this sounds like a hack. "Things get screwy when you screw with them, UNLESSSSS we add a magic thing that undoes our work". Well yeah.
nextaccountic 5y ago
I don't know about quantum mechanics, but when we talk about space we should be free to add a quantity to the whole universe (like adding 1 to the x coordinate of everything) because this just shifts the whole universe, or accordingly, shifts the origin - the (0, 0, 0) point - in the opposite direction. The origin is set at an arbitrary point so this "space shift invariance" is saying that it doesn't matter what point we set for the origin (and mathematically this corresponds to the conservation of momentum - see Noether's theorem[0]) Hmm maybe the "zero" for the quantum states is arbitrary, so you should be able to add anything to it for the whole universe, and this merely changes the zero state in the opposite direction.. and since this should be a conversation law, pretty sure this is equivalent to the conservation of electric charge https://en.wikipedia.org/wiki/Noether's_theorem https://en.wikipedia.org/wiki/Noether's_theorem
- galaxyLogic 5y ago
  I may be starting to get it. The magic numbers we add to everything or multiply everything with really represents just a change in the viewpoint, origin of the observer. And because it should be possible to change the "location" of the observer (say the voltage we take to be 0 volts) and still get the same theory to hold up, we can discover that it can only hold up if we assume the existence of some new field. Something like this?
  - layoutIfNeeded 5y ago
    Yes.
- nextaccountic 5y ago
  *conservation law

ufo_pilot 5y ago

Not GP, or a physicist, but my understanding is that the different number at each point you multiply with represents a degree of freedom at each point of spacetime, and in this degree of freedom is where the electromagnetic field lives.

tagrun 5y ago

To find out what kind of symmetry a field (as described by a Lagrangian) admits. It's called gauge symmetry: https://en.wikipedia.org/wiki/Gauge_theory https://en.wikipedia.org/wiki/Gauge_theory It's a complex number for QED but in general, it's a unitary matrix: a rotation which preserves the magnitude of a wavefunction (a complex vector). In physics, symmetries play a fundamental role.

cshimmin 5y ago

The simple answer is "because we can". In general, physicists have found that we should write down the most general mathematical theory compatible with what's observed. A famous example of this is Einstein's cosmological constant -- I'll leave that one to wikipedia [1] since I'm a particle physicist and not an expert on GR. In the case of gauge theory, the idea that we should consider the most general case has been well proven. As GP pointed out, all observable phenomena ultimately depend only on the absolute modulus of the field Ψ, so a theoretical physicist naturally wonders, what happens if you allow its complex phase to vary. Turns out nothing interesting happens if you apply a global phase, but if you allow the phase to vary at every point in spacetime, it ends up breaking the theory. That is, unless you include an additional field at every point in spacetime that precisely cancels out the change induced by the gauge freedom. In other words, the motivation is that we can't simply look and "see" whether or not there is a locally varying phase on the wavefunction Ψ, since we only can measure |Ψ|^2. So we have to assume there is, until proven otherwise. Since a local phase would imply the existence of an extra field to cancel it out, we can indirectly check for this scenario by looking for the corresponding field. As pointed out by GP, in the case of a U(1) gauge, it turns out there is such a field, and electromagnetism (and all of its laws) exactly fit the bill. There are other "unmeasurable" symmetries you could apply to the wave function as well, beyond just a complex phase. SU(2) is a Lie group symmetry which would mean that the measurable properties of certain tuples of fields (Ψ, ϕ) are indistinguishable under a sort of complex-valued rotation of Ψ->ϕ and ϕ->Ψ. Again, if you assume a such symmetry is locally varied at every point in spacetime, you end up requiring not one but three new fields to cancel out the effects on the SM Lagrangian. It turns out that the vector bosons W+, W-, and Z, which mediate weak nuclear forces exactly fit the bill. [1] - https://en.wikipedia.org/wiki/Cosmological_constant https://en.wikipedia.org/wiki/Cosmological_constant

cominous 5y ago
Wow you blew my mind. You need to consider creating YouTube videos about this.
cshimmin 5y ago
And at the risk of muddying the original point, it turns out that strictly speaking the neither the neutral Z boson nor the electromagnetic photon can be said to come from SU(2) or U(1). Instead, each field is a linear combination of some abstract fields (namely the neutral W0 and the weak hypercharge B bosons) from the unitary gauge induced by the compound symmetry SU(2)xU(1). This is because when both symmetries SU(2) and U(1) are present, there are different ways you can "mix" the two into the Lagrangian. The mixing that gives us the photon and the Z0 boson is known because of the experimental confirmation of these particles. This is what is meant when it is said that electromagnetism and weak neuclear forces are united in a higher-energy theory as a single "electroweak" force.
- trenchgun 5y ago
  Thank you!
- jasonwatkinspdx 5y ago
  Thanks for these comments. This simplified a whole bunch of things I had questions about.
- tinco 5y ago
  This is great, you perfectly bridged the gap between having been 25 once and vaguely remembering the math terms mentioned and bringing it back to reality. Really epic how this (at least to me) abstract and arcane math not only explains the behaviors of the subatomic particles, but even predicts the existence of more and their related forces. The work of theoretical physicists always seemed like the other side of an ocean of math away, but thanks to this explanation it feels like I can at least make out a lighthouse in the distance.
vehementi 5y ago
> That is, unless you include an additional field at every point in spacetime that precisely cancels out the change induced by the gauge freedom. Could you elaborate on this a bit? To a layperson this sounds like a hack. "Things get screwy when you screw with them, UNLESSSSS we add a magic thing that undoes our work". Well yeah.

nextaccountic 5y ago

I don't know about quantum mechanics, but when we talk about space we should be free to add a quantity to the whole universe (like adding 1 to the x coordinate of everything) because this just shifts the whole universe, or accordingly, shifts the origin - the (0, 0, 0) point - in the opposite direction. The origin is set at an arbitrary point so this "space shift invariance" is saying that it doesn't matter what point we set for the origin (and mathematically this corresponds to the conservation of momentum - see Noether's theorem[0]) Hmm maybe the "zero" for the quantum states is arbitrary, so you should be able to add anything to it for the whole universe, and this merely changes the zero state in the opposite direction.. and since this should be a conversation law, pretty sure this is equivalent to the conservation of electric charge https://en.wikipedia.org/wiki/Noether's_theorem https://en.wikipedia.org/wiki/Noether's_theorem

galaxyLogic 5y ago
I may be starting to get it. The magic numbers we add to everything or multiply everything with really represents just a change in the viewpoint, origin of the observer. And because it should be possible to change the "location" of the observer (say the voltage we take to be 0 volts) and still get the same theory to hold up, we can discover that it can only hold up if we assume the existence of some new field. Something like this?
- layoutIfNeeded 5y ago
  Yes.
nextaccountic 5y ago
*conservation law