# Quantum Mechanics in 1+1 spacetime

I was trying to make a problem simpler by working in 1+1 spacetime, and I realized that it's far from obvious that quantum physics would even work in this case. Any non-spin related phenomenon could still work (e.g. quantum scalar fields, schrodinger equation) but in less than 2 spatial dimensions angular momentum has no real definition, and in principle shouldn't even be possible. Fermions should be impossible, but no equation comes to mind where I can pinpoint why it breaks down in only 1 spatial dimension (e.g. the Dirac equation still works fine in 1+1).

Another issue I have is the role of Planck's constant throughout all of Quantum physics. This constant is very closely related to angular momentum (even though it could just be considered distance*momentum), so in 1+1 dimension is there any quantum phenomenon that still works? Clearly isolated parts of quantum physics, like the schrodinger equation, would not break down on theoretical grounds. However, I'm wondering if something more fundamental such as QFT would fail in the scalar case where Planck's constant will still appear.

I'm a little tired at the moment, but I think what I'm basically looking for an answer to is:
1) Am I correct that no analogue to angular momentum is possible in a 1+1 spacetime?
2) How does the breakdown of angular momentum affect quantum mechanics, given how closely the two are related?

Simon Bridge
Homework Helper
The usual 1+1 simplification includes some reason to ignore the other dimensions.
i.e. maybe the other two space dimensions are not restricted compared with the dimension of interest.

Some notes:
In QM, angular momentum can be an intrinsic property - certainly spin is usually treated that way.
Orbital angular momentum would be a tad trickier.

Rather than figuring what "should" be - try working out what actually is.

The usual 1+1 simplification includes some reason to ignore the other dimensions.
i.e. maybe the other two space dimensions are not restricted compared with the dimension of interest.

Some notes:
In QM, angular momentum can be an intrinsic property - certainly spin is usually treated that way.
Orbital angular momentum would be a tad trickier.

Rather than figuring what "should" be - try working out what actually is.

Yes, I'm past the 1+1 simplification and I'm just curious about what QFT in a 1+1 universe would be like, where the other dimensions just aren't there. I know spin is an intrinsic property, but it is still a form angular momentum (e.g. J=S+L is the conserved quantity). There are no mechanical reasons it can't exist like for orbital angular momentum, but the entire concept of angular momentum is ill-defined with only one spatial dimension.

What I'm doing is very common in physics: I'm going into the problem with an idea of what the answer should be based on my intuition, and then seeing if the math agrees. I'm not forcing it to agree or assuming it must agree, but I have some qualitative arguments why QM should face serious issues in 1+1 dimensions. My question is whether or not this is supported by the math, and my confusion is about my inability to find anything similar (i.e. I can't think of any quantum equation that isn't easily extended to a 1+1 spacetime). However, like you said 1+1 is usually just a restriction on our 3+1 world, so it's possible that what I think is an equation for a 1+1 quantum world actually depends on additional dimensions in some obscured way. I think the only real way to do this is to look at a general 1+1 QFT and see if it becomes inconsistent or trivial...

Simon Bridge
Homework Helper
Yes, I'm past the 1+1 simplification and I'm just curious about what QFT in a 1+1 universe would be like, where the other dimensions just aren't there.
... so this is just empty speculation?

I know spin is an intrinsic property, but it is still a form angular momentum (e.g. J=S+L is the conserved quantity). There are no mechanical reasons it can't exist like for orbital angular momentum, but the entire concept of angular momentum is ill-defined with only one spatial dimension.
You mean because physical rotations don't make sense? That's just the analogy that gives us the name - the spin with only one available spacial dimension would not have all the effects of spin with 3.

What I'm doing is very common in physics: I'm going into the problem with an idea of what the answer should be based on my intuition, and then seeing if the math agrees.
... so, logically, your next step is to do the maths and see what it actually says. You have yet to show us any.
The usual approach is to attempt to disprove the prior assumption.

Note: a mathematical model does not need to be restricted to the space-time dimensions available - see string theory.
Perhaps you could look at how magnets would work in 1D as a kickoff? Would the magnetic moment be restricted to the available space dimension?

Anyway - If we go too far the wrong way we'll brush up against the "no speculation" rule.

... so this is just empty speculation?
It's not speculation... I'm trying to see what things would look like in fewer dimensions. Plenty of people generalize theories to more than 3+1 dimensions, I just want to take that generalization and look at 1+1

You mean because physical rotations don't make sense? That's just the analogy that gives us the name - the spin with only one available spacial dimension would not have all the effects of spin with 3.
It isn't just an analogy, angular momentum is the conserved quantity associated with invariance under rotations and spin is a form of angular momentum. If you have no rotations, angular momentum as a conserved quantity makes no sense. Further, the n-dimensional generalization of angular momentum is the antisymmetry tensor product of position and momentum. In 2+1 the spatial part of this tensor (angular momentum) is a scalar, and in 1+1 it just doesn't exist

... so, logically, your next step is to do the maths and see what it actually says. You have yet to show us any.
The usual approach is to attempt to disprove the prior assumption.
If I had done this I wouldn't be here. I came here to ask if anyone had already done the math or knew the answer, because I'm not sure how to prove or disprove the assumption

Note: a mathematical model does not need to be restricted to the space-time dimensions available - see string theory.
Perhaps you could look at how magnets would work in 1D as a kickoff? Would the magnetic moment be restricted to the available space dimension?

Anyway - If we go too far the wrong way we'll brush up against the "no speculation" rule.
Any particular string theory has a well-defined number of dimensions, they're just compactified. The exact method people take to generalize our 3+1 dimensional spacetime into n+1 dimensional spacetime is what I'm trying to use to look at 1+1. Also, magnets would not exist in 1D. This can easily be seen by looking at the covariant form of Maxwell's equations. The 2x2 field tensor is antisymmetric, so it only has 1 free parameter: the electric field

Fredrik
Staff Emeritus
Gold Member
I'm a little tired at the moment, but I think what I'm basically looking for an answer to is:
1) Am I correct that no analogue to angular momentum is possible in a 1+1 spacetime?
2) How does the breakdown of angular momentum affect quantum mechanics, given how closely the two are related?
1. Different irreducible representations of the symmetry group of spacetime correspond to different particle species, so changing the spacetime would change the symmetry group, and therefore change what particles can be described. Since the 1+1-dimensional Poincaré group doesn't include rotations, spin would no longer be one of the numbers that label different particle species. In other words, there wouldn't be such a thing as spin.

2. Not sure what you mean. If you're asking what particles can exist in a 1+1-dimensional spacetime, I don't have a complete answer for you. I think the only spacetime-related quantity that would label a particle species is mass. But different particle species can also be distinguished by numbers that have nothing to do with the underlying spacetime.

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1. Different irreducible representations of the symmetry group of spacetime correspond to different particle species, so changing the spacetime would change the symmetry group, and therefore change what particles can be described. Since the 1+1-dimensional Poincaré group doesn't include rotations, spin would no longer be one of the numbers that label different particle species. In other words, there wouldn't be such a thing as spin.

2. Not sure what you mean. If you're asking what particles can exist in a 1+1-dimensional spacetime, I don't have a complete answer for you. I think the only spacetime-related quantity that would label a particle species is mass. But different particle species can also be distinguished by numbers that have nothing to do with the underlying spacetime.

Thanks Fredrik, that does answer my first question. For the main question though, I'm referring to the the role angular momentum plays in basic single-particle quantum mechanics. In QFT you're right, the dependence on angular momentum is kindof obscured, and you could argue Planck's constant just relates frequencies to energy. So the question would just be "what fields can exist" and the answer would probably be something along the lines of "only scalar fields" (my guess). For the quantum phenomena that predate QFT though (e.g. heisenberg uncertainty principle, schrodinger/klein-gordan/dirac equations, ...) angular momentum seems to play a very fundamental role, since it is the only quantity which is always quantized. The quantum "scale" is usually parameterized with Planck's constant, so it seems strange in a case where angular momentum can not exist to use a fundamental constant that represents an angular momentum

*edit* Also, I am curious what fields can exist in 1+1 spacetime if you have an answer to that. I wouldn't be surprised if vector and spinor fields were prohibited, but the scalar case seems interesting (i.e. what are the conditions of renormalizability? are interactions allowed or is it a trivial theory? what types of interactions are allowed?)

Spinnor
Gold Member
...
I'm a little tired at the moment, but I think what I'm basically looking for an answer to is:
1) Am I correct that no analogue to angular momentum is possible in a 1+1 spacetime?
...

Yes and No. I believe interesting QM stuff happens in 1+1 but no, spin as we "know" it does not exist (but keeping an open mind). From a little studying on the matter I have come up with some clues and I think your interest is well motivated.

The Dirac equation in 1+1 has spinor solutions, see equation 2 in,

http://home.pcisys.net/~bestwork.1/QQM/dimension_dirac.htm

See some interesting facts about spinors in 1+1, open link and page down to table summary,

http://en.wikipedia.org/wiki/Spinor#Summary_in_low_dimensions

In particular note

Metric signature, left-handed Weyl, right-handed Weyl, conjugacy, Dirac left-handed, Majorana–Weyl right-handed, Majorana–Weyl, Majorana

complex complex complex real real real
(2,0) 1 1 mutual 2 – – 2
(1,1) 1 1 self 2 1 1 2
(3,0) – – – 2 – – –
(2,1) – – – 2 – – 2
(4,0) 2 2 self 4 – – –
(3,1) 2 2 mutual 4 – – 4

Lots of interesting stuff that we associate spin and spinors with occur in 1+1, what this "means", lets find out.

Second clue that you might be right comes from a foot note in a String Theory book,

Bottom of page 187, https://books.google.com/books?id=_18hAwAAQBAJ&pg=PA187&dq=string theory: volume 1, introduction&hl=en&sa=X&ei=dr-yVOTQO8rasAStkIGQBg&ved=0CCwQ6AEwAw#v=onepage&q=string theory: volume 1, introduction&f=false

"Properly speaking, in 1+1 dimensions there is no such thing as spin, but there is a two-dimensional Lorentz group (or local Lorentz group, in the case of a generally covariant theory), and it makes sense to ask how a two-dimensional field transforms under this group. The spin-statistics theorem says that in local quantum field theory in two dimensions an anticommuting field must have half-integral Lorentz quantum numbers."

The "half-integral Lorentz quantum numbers" I take to be spinor representations of the Lorentz group in 1+1(hope I worded that right?). The group must be pretty small but not trivial?

Another clue, only in odd space dimension(s) can handedness be defined, I think that applies to 1+1 what ever that means.

Maybe we can work together and figure out this stuff.

Thanks for any help!

Hmm I'm a little confused now. You're right that the Dirac equation does have 1+1 solutions, and there should be some fermion-like field possible because there are still 2 dimensions. However, here are my issues:
1) The lowest dimensional Dirac equation solutions only have 2 components. I haven't worked out the specifics yet, but I would assume these correspond to positive/negative energy fields with no chirality. The thing is, in a world without spin, how can you describe fermions which must have spin by definition? I'm wondering if its possible to show that the solutions to the 1+1 Dirac equation are not fermionic?
2) Fermions can't exist, but as you said this is still a 2 dimensional world. So there should be some spin-like property that behaves nothing like spin, and particles with 1/2 "spin" could be considered fermions. However, what is the 3+1 analog of this?? You would need some conserved quantity associated with "rotations" around space-time planes. I believe this is called "mass moment", and it is a conserved quantity. However, I have NEVER heard of anyone trying to quantize this...
http://en.wikipedia.org/wiki/Relativistic_angular_momentum#Dynamic_mass_moment
N=Ex-tp is the mass moment in 3+1 spacetime, but I've never worked with it in quantum mechanics (although I will take a closer look tonight)

Spinnor
Gold Member
I wonder if the charge of these imaginary electrons in 1+1 needs to be considered as well? Slowly working on this.

The math for the 3+1 D Dirac equation is set up nicely in the following, \$ 5.92 + shipping. Just simplify for 1+1.

https://www.amazon.com/gp/product/0852743289/?tag=pfamazon01-20 Spinnor
Gold Member
The following paper goes some detail that might help with understanding the Dirac equation in 1+1, see section 1.3
...
The word “chiral” comes from the Greek word for hand, χιρ. The projection
operators P+ and P− are often called PR and PL respectively; what does handedness
have to do with the matrix Γ? Consider the Lagrangian for a free massive Dirac fermion
in 1 + 1 dimensions ...

http://arxiv.org/pdf/0912.2560.pdf

edit, and section 2,

...

2.1 The U(1)A anomaly in 1+1 dimensions
One of the fascinating features of chiral symmetry is that sometimes it is not a symmetry
of the quantum field theory even when it is a symmetry of the Lagrangian. In
particular, Noether’s theorem can be modified in a theory with an infinite number of
degrees of freedom; the modification is called “an anomaly”. Anomalies turn out to
be very relevant both for phenomenology, and for the implementation of lattice field
theory. The reason anomalies affect chiral symmetries is that regularization requires a
cut-off on the infinite number of modes above some mass scale, while chiral symmetry
is incompatible with fermion masses.

Anomalies can be seen in many different ways. I think the most physical is to
look at what happens to the ground state of a theory with a single flavor of massless
Dirac fermion in (1 + 1) dimensions in the presence of an electric field. Suppose one
adiabatically turns on a constant positive electric field E(t), then later turns it off; the
equation of motion for the fermion is 2 dp
dt = eE(t) and the total change in momentum
is
...

All very complex, and we are only in 1+1.

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Ah that's interesting. So it appears that chirality really has nothing to do with spin, even though it is related to helicity in 3+1 spacetime. It also seems that fermions don't even have chirality in 2+1 spacetime, despite the possibility of spin (although helicity is still impossible). The way this guy explains it makes it clearer that chirality is just a manifestation of the reducibility of even dimensional spacetimes.

So fermions are possible in 1+1 dimensions, but what happens to spin? In basic quantum mechanics you learn that bosons have integer spin and fermions have half integer spin. I'm not sure how this fact is derived from QFT, so I have no idea how it can be generalized to other spacetimes

Spinnor
Gold Member
In basic quantum mechanics you learn that bosons have integer spin and fermions have half integer spin. I'm not sure how this fact is derived from QFT, so I have no idea how it can be generalized to other spacetimes

The book I suggested above goes into this on pages 118 to 120. Google "spin statistics theorem" and lots comes up. Do a Google book search of the same, "spin statistics theorem" and Google will usually present several pages of good "red meat". Trying to understand.

Edit, I forgot to suggest a Physics Today article by Nobel Prize winner David Gross that also gets to your question. Here,