Does the CPT theorem hold for all spacetime dimensions?What makes *electric* charge special (wrt. CPT theorem)?Vanishing of the Ricci tensor in higher spacetime dimensionsWhat feature of QFT requires the C in the CPT theorem?Is it possible for there to be a spacetime which shares one of our dimensions?Is it possible to measure the number of spacetime dimensions?Spacetime dimension and the dimension of Clifford algebraWhat is the spin-statistics theorem in higher dimensions?Interacting conformal field theories in spacetime dimensions higher 6?
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Does the CPT theorem hold for all spacetime dimensions?
What makes *electric* charge special (wrt. CPT theorem)?Vanishing of the Ricci tensor in higher spacetime dimensionsWhat feature of QFT requires the C in the CPT theorem?Is it possible for there to be a spacetime which shares one of our dimensions?Is it possible to measure the number of spacetime dimensions?Spacetime dimension and the dimension of Clifford algebraWhat is the spin-statistics theorem in higher dimensions?Interacting conformal field theories in spacetime dimensions higher 6?
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I can't find any reference which mentions the dependence of the theorem on spacetime dimension, but it would be very interesting to know what if any it has!
quantum-field-theory special-relativity spacetime-dimensions cpt-symmetry
edited Jun 27 at 1:47
Qmechanic♦
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I can't find any reference which mentions the dependence of the theorem on spacetime dimension, but it would be very interesting to know what if any it has!
quantum-field-theory special-relativity spacetime-dimensions cpt-symmetry
edited Jun 27 at 1:47
Qmechanic♦
116k14 gold badges230 silver badges1382 bronze badges
asked Jun 26 at 23:24
LucashWindowWasherLucashWindowWasher
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$endgroup$
add a comment
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$begingroup$
I can't find any reference which mentions the dependence of the theorem on spacetime dimension, but it would be very interesting to know what if any it has!
quantum-field-theory special-relativity spacetime-dimensions cpt-symmetry
edited Jun 27 at 1:47
Qmechanic♦
116k14 gold badges230 silver badges1382 bronze badges
asked Jun 26 at 23:24
LucashWindowWasherLucashWindowWasher
5272 silver badges15 bronze badges
$endgroup$
I can't find any reference which mentions the dependence of the theorem on spacetime dimension, but it would be very interesting to know what if any it has!
quantum-field-theory special-relativity spacetime-dimensions cpt-symmetry
quantum-field-theory special-relativity spacetime-dimensions cpt-symmetry
edited Jun 27 at 1:47
Qmechanic♦
116k14 gold badges230 silver badges1382 bronze badges
asked Jun 26 at 23:24
LucashWindowWasherLucashWindowWasher
5272 silver badges15 bronze badges
edited Jun 27 at 1:47
Qmechanic♦
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asked Jun 26 at 23:24
LucashWindowWasherLucashWindowWasher
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edited Jun 27 at 1:47
Qmechanic♦
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edited Jun 27 at 1:47
Qmechanic♦
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edited Jun 27 at 1:47
Qmechanic♦
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116k14 gold badges230 silver badges1382 bronze badges
asked Jun 26 at 23:24
LucashWindowWasherLucashWindowWasher
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asked Jun 26 at 23:24
LucashWindowWasherLucashWindowWasher
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asked Jun 26 at 23:24
LucashWindowWasherLucashWindowWasher
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add a comment
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1 Answer
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With one minor qualification, the answer is yes: the CPT theorem holds for all spacetime dimensions.
The qualification is that the P in CPT should be interpreted as a reflection of an odd number of spatial dimensions. (The simplest choice is to reflect just one spatial dimension.) If the total number of spatial dimensions is odd, then this is the same as reflecting all of them; but if the total number of spatial dimensions is even, the distinction is important. That's because a reflection of an even number of spatial dimensions can be un-done by ordinary rotations that are continuously connected to the identity. The P in CPT should not be continuously connected to the identity.
To address this, Witten suggests using a new symbol $R$ for a reflection of a single spatial dimension, so the CPT theorem would be called the CRT theorem. Here's an excerpt from page 5 in "Fermion Path Integrals And Topological Phases", http://arxiv.org/abs/1508.04715, where $D$ denotes the number of spacetime dimensions:
...symmetry that in $D = 4$ is usually called CPT (where C is charge
conjugation and P is parity or spatial inversion). However, this
formulation is not valid for odd $D$ (even spatial dimension $d$), since
in that case P is contained in the connected part of the spatial
rotation group and should be replaced by an operation that reverses
the orientation of space. To use a language that is equivalent to the
usual CPT statement for even $D$ but is uniformly valid for all $D$, we
will refer instead to CRT symmetry, where R is a reflection of one
spatial coordinate. CRT symmetry is always valid in a relativistic
theory of any dimension...
Some differences between the effect of CPT in $4k$ and $4k+2$ spacetime dimensions are mentioned in section 4 of Alvarez-Gaumé and Witten (1983), "Gravitational anomalies," Nuclear Physics B 234: 269-330.
answered Jun 26 at 23:37
Chiral AnomalyChiral Anomaly
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$endgroup$
1
$begingroup$
Thank you! This question came out of my curiosity for how a 3d dirac field transforms under parity, so your reference was perfect.
$endgroup$
– LucashWindowWasher
Jun 27 at 23:02
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1 Answer
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1 Answer
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$begingroup$
With one minor qualification, the answer is yes: the CPT theorem holds for all spacetime dimensions.
The qualification is that the P in CPT should be interpreted as a reflection of an odd number of spatial dimensions. (The simplest choice is to reflect just one spatial dimension.) If the total number of spatial dimensions is odd, then this is the same as reflecting all of them; but if the total number of spatial dimensions is even, the distinction is important. That's because a reflection of an even number of spatial dimensions can be un-done by ordinary rotations that are continuously connected to the identity. The P in CPT should not be continuously connected to the identity.
To address this, Witten suggests using a new symbol $R$ for a reflection of a single spatial dimension, so the CPT theorem would be called the CRT theorem. Here's an excerpt from page 5 in "Fermion Path Integrals And Topological Phases", http://arxiv.org/abs/1508.04715, where $D$ denotes the number of spacetime dimensions:
...symmetry that in $D = 4$ is usually called CPT (where C is charge
conjugation and P is parity or spatial inversion). However, this
formulation is not valid for odd $D$ (even spatial dimension $d$), since
in that case P is contained in the connected part of the spatial
rotation group and should be replaced by an operation that reverses
the orientation of space. To use a language that is equivalent to the
usual CPT statement for even $D$ but is uniformly valid for all $D$, we
will refer instead to CRT symmetry, where R is a reflection of one
spatial coordinate. CRT symmetry is always valid in a relativistic
theory of any dimension...
Some differences between the effect of CPT in $4k$ and $4k+2$ spacetime dimensions are mentioned in section 4 of Alvarez-Gaumé and Witten (1983), "Gravitational anomalies," Nuclear Physics B 234: 269-330.
answered Jun 26 at 23:37
Chiral AnomalyChiral Anomaly
21.3k4 gold badges29 silver badges65 bronze badges
$endgroup$
1
$begingroup$
Thank you! This question came out of my curiosity for how a 3d dirac field transforms under parity, so your reference was perfect.
$endgroup$
– LucashWindowWasher
Jun 27 at 23:02
add a comment
|
$begingroup$
With one minor qualification, the answer is yes: the CPT theorem holds for all spacetime dimensions.
The qualification is that the P in CPT should be interpreted as a reflection of an odd number of spatial dimensions. (The simplest choice is to reflect just one spatial dimension.) If the total number of spatial dimensions is odd, then this is the same as reflecting all of them; but if the total number of spatial dimensions is even, the distinction is important. That's because a reflection of an even number of spatial dimensions can be un-done by ordinary rotations that are continuously connected to the identity. The P in CPT should not be continuously connected to the identity.
To address this, Witten suggests using a new symbol $R$ for a reflection of a single spatial dimension, so the CPT theorem would be called the CRT theorem. Here's an excerpt from page 5 in "Fermion Path Integrals And Topological Phases", http://arxiv.org/abs/1508.04715, where $D$ denotes the number of spacetime dimensions:
...symmetry that in $D = 4$ is usually called CPT (where C is charge
conjugation and P is parity or spatial inversion). However, this
formulation is not valid for odd $D$ (even spatial dimension $d$), since
in that case P is contained in the connected part of the spatial
rotation group and should be replaced by an operation that reverses
the orientation of space. To use a language that is equivalent to the
usual CPT statement for even $D$ but is uniformly valid for all $D$, we
will refer instead to CRT symmetry, where R is a reflection of one
spatial coordinate. CRT symmetry is always valid in a relativistic
theory of any dimension...
Some differences between the effect of CPT in $4k$ and $4k+2$ spacetime dimensions are mentioned in section 4 of Alvarez-Gaumé and Witten (1983), "Gravitational anomalies," Nuclear Physics B 234: 269-330.
answered Jun 26 at 23:37
Chiral AnomalyChiral Anomaly
21.3k4 gold badges29 silver badges65 bronze badges
$endgroup$
1
$begingroup$
Thank you! This question came out of my curiosity for how a 3d dirac field transforms under parity, so your reference was perfect.
$endgroup$
– LucashWindowWasher
Jun 27 at 23:02
add a comment
|
$begingroup$
With one minor qualification, the answer is yes: the CPT theorem holds for all spacetime dimensions.
The qualification is that the P in CPT should be interpreted as a reflection of an odd number of spatial dimensions. (The simplest choice is to reflect just one spatial dimension.) If the total number of spatial dimensions is odd, then this is the same as reflecting all of them; but if the total number of spatial dimensions is even, the distinction is important. That's because a reflection of an even number of spatial dimensions can be un-done by ordinary rotations that are continuously connected to the identity. The P in CPT should not be continuously connected to the identity.
To address this, Witten suggests using a new symbol $R$ for a reflection of a single spatial dimension, so the CPT theorem would be called the CRT theorem. Here's an excerpt from page 5 in "Fermion Path Integrals And Topological Phases", http://arxiv.org/abs/1508.04715, where $D$ denotes the number of spacetime dimensions:
...symmetry that in $D = 4$ is usually called CPT (where C is charge
conjugation and P is parity or spatial inversion). However, this
formulation is not valid for odd $D$ (even spatial dimension $d$), since
in that case P is contained in the connected part of the spatial
rotation group and should be replaced by an operation that reverses
the orientation of space. To use a language that is equivalent to the
usual CPT statement for even $D$ but is uniformly valid for all $D$, we
will refer instead to CRT symmetry, where R is a reflection of one
spatial coordinate. CRT symmetry is always valid in a relativistic
theory of any dimension...
Some differences between the effect of CPT in $4k$ and $4k+2$ spacetime dimensions are mentioned in section 4 of Alvarez-Gaumé and Witten (1983), "Gravitational anomalies," Nuclear Physics B 234: 269-330.
answered Jun 26 at 23:37
Chiral AnomalyChiral Anomaly
21.3k4 gold badges29 silver badges65 bronze badges
$endgroup$
With one minor qualification, the answer is yes: the CPT theorem holds for all spacetime dimensions.
The qualification is that the P in CPT should be interpreted as a reflection of an odd number of spatial dimensions. (The simplest choice is to reflect just one spatial dimension.) If the total number of spatial dimensions is odd, then this is the same as reflecting all of them; but if the total number of spatial dimensions is even, the distinction is important. That's because a reflection of an even number of spatial dimensions can be un-done by ordinary rotations that are continuously connected to the identity. The P in CPT should not be continuously connected to the identity.
To address this, Witten suggests using a new symbol $R$ for a reflection of a single spatial dimension, so the CPT theorem would be called the CRT theorem. Here's an excerpt from page 5 in "Fermion Path Integrals And Topological Phases", http://arxiv.org/abs/1508.04715, where $D$ denotes the number of spacetime dimensions:
...symmetry that in $D = 4$ is usually called CPT (where C is charge
conjugation and P is parity or spatial inversion). However, this
formulation is not valid for odd $D$ (even spatial dimension $d$), since
in that case P is contained in the connected part of the spatial
rotation group and should be replaced by an operation that reverses
the orientation of space. To use a language that is equivalent to the
usual CPT statement for even $D$ but is uniformly valid for all $D$, we
will refer instead to CRT symmetry, where R is a reflection of one
spatial coordinate. CRT symmetry is always valid in a relativistic
theory of any dimension...
Some differences between the effect of CPT in $4k$ and $4k+2$ spacetime dimensions are mentioned in section 4 of Alvarez-Gaumé and Witten (1983), "Gravitational anomalies," Nuclear Physics B 234: 269-330.
answered Jun 26 at 23:37
Chiral AnomalyChiral Anomaly
21.3k4 gold badges29 silver badges65 bronze badges
edited Jun 26 at 23:44
answered Jun 26 at 23:37
Chiral AnomalyChiral Anomaly
21.3k4 gold badges29 silver badges65 bronze badges
answered Jun 26 at 23:37
Chiral AnomalyChiral Anomaly
21.3k4 gold badges29 silver badges65 bronze badges
answered Jun 26 at 23:37
Chiral AnomalyChiral Anomaly
21.3k4 gold badges29 silver badges65 bronze badges
21.3k4 gold badges29 silver badges65 bronze badges
1
$begingroup$
Thank you! This question came out of my curiosity for how a 3d dirac field transforms under parity, so your reference was perfect.
$endgroup$
– LucashWindowWasher
Jun 27 at 23:02
add a comment
|
1
$begingroup$
Thank you! This question came out of my curiosity for how a 3d dirac field transforms under parity, so your reference was perfect.
$endgroup$
– LucashWindowWasher
Jun 27 at 23:02
1
1
$begingroup$
Thank you! This question came out of my curiosity for how a 3d dirac field transforms under parity, so your reference was perfect.
$endgroup$
– LucashWindowWasher
Jun 27 at 23:02
$begingroup$
Thank you! This question came out of my curiosity for how a 3d dirac field transforms under parity, so your reference was perfect.
$endgroup$
– LucashWindowWasher
Jun 27 at 23:02
add a comment
|
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