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1. There are already many good answers explaining the conventional theory and observations. Nevertheless, related to comments by lurscher and Adam Zalcman, it seems appropriate to mention the Nambu bracket, which is a Poisson-like bracket

$$ \{ f,g,h \} $$

with 3 function entries, originally invented by Nambu in 1973, purportedly in a failed attempt to explain $SU(3)$ symmetry in quarks. Already Nambu discusses an operator version

$$ [\hat{f},\hat{g},\hat{h} ], $$

and one can imagine some kind of uncertainty relation associated to this, where canonical variables come in triplets. Unfortunately, the subject so far has remained just theoretical speculations. [Authors even don't agree what should replace the Jacobi identity for the Poisson bracket $\{ f,g\}$, although most think it should be the so-called Filippov fundamental Identity (FI).]

2. Recently in 2008, the Nambu-bracket has been used in the Bagger–Lambert–Gustavsson M2 brane proposal.

3. The exist various generalizations to higher Nambu brackets

$$ \{ f_1,\ldots ,f_n \} $$

with $n$ entries.

4. For more information, see this review.

1. There are already many good answers explaining the conventional theory and observations. Nevertheless, related to comments by lurscher and Adam Zalcman, it seems appropriate to mention the Nambu bracket, which is a Poisson-like bracket

$$ \{ f,g,h \} $$

with 3 function entries, originally invented by Nambu in 1973, purportedly in a failed attempt to explain the $SU(3)$ symmetry of quarks. Already Nambu discusses an operator 3-bracket

$$ [\hat{f},\hat{g},\hat{h}], $$

and one can imagine some kind of uncertainty relation associated to this, where canonical variables come in triples. Unfortunately, the subject so far has remained just theoretical speculations. [Authors even don't agree what should replace the Jacobi identity for the Poisson bracket $\{ f,g\}$, although most think it should be the so-called Filippov fundamental Identity (FI).]

2. Recently in 2008, the Nambu-bracket has been used in the Bagger–Lambert–Gustavsson M2 brane proposal.

3. The exist various generalizations to higher Nambu brackets

$$ \{ f_1,\ldots ,f_n \} $$

with $n$ entries.

4. For more information, see this recent review.

1. There are already many good answers. Nevertheless, it seems appropriate in this connection to mention the Nambu bracket, which is a Poisson-like bracket

$$ \{ f,g,h \} $$

with 3 function entries, originally invented by Nambu in 1973, purportedly in a failed attempt to explain $SU(3)$ symmetry in quarks. Already Nambu discusses an operator version

$$ [\hat{f},\hat{g},\hat{h} ], $$

and one can imagine some kind of uncertainty relation associated to this, where canonical variables come in triplets. Unfortunately, the subject so far has remained just theoretical speculations. [Authors even don't agree what should replace the Jacobi identity for the Poisson bracket $\{ f,g\}$, although most think it should be the so-called Filippov fundamental Identity (FI).]

2. Recently in 2008, Nambu-brackets have been discussed in the Bagger–Lambert–Gustavsson M2 brane proposal.

3. The exist various generalizations to higher Nambu brackets

$$ \{ f_1,\ldots ,f_n \} $$

with $n$ entries.

4. For more information, see this review.

1. There are already many good answers explaining the conventional theory and observations. Nevertheless, related to comments by lurscher and Adam Zalcman, it seems appropriate to mention the Nambu bracket, which is a Poisson-like bracket

$$ \{ f,g,h \} $$

with 3 function entries, originally invented by Nambu in 1973, purportedly in a failed attempt to explain $SU(3)$ symmetry in quarks. Already Nambu discusses an operator version

$$ [\hat{f},\hat{g},\hat{h} ], $$

and one can imagine some kind of uncertainty relation associated to this, where canonical variables come in triplets. Unfortunately, the subject so far has remained just theoretical speculations. [Authors even don't agree what should replace the Jacobi identity for the Poisson bracket $\{ f,g\}$, although most think it should be the so-called Filippov fundamental Identity (FI).]

2. Recently in 2008, the Nambu-bracket has been used in the Bagger–Lambert–Gustavsson M2 brane proposal.

3. The exist various generalizations to higher Nambu brackets

$$ \{ f_1,\ldots ,f_n \} $$

with $n$ entries.

4. For more information, see this review.

1. It seems appropriate to mention the Nambu bracket, which is a Poisson-like bracket

$$ \{ f,g,h \} $$

with 3 function entries, originally invented by Nambu in 1973, purportedly in a failed attempt to explain $SU(3)$ symmetry in quarks. Already Nambu discusses an operator version

$$ [\hat{f},\hat{g},\hat{h} ], $$

and one can imagine some kind of uncertainty relation associated to this, where canonical variables come in triplets. It is fair to say, that the subject remains speculative. [Authors even don't agree what should replace the Jacobi identity for the Poisson bracket $\{ f,g\}$, although most think it should be the so-called Filippov fundamental Identity (FI).]

2. Recently in 2008, Nambu-brackets have been discussed in the Bagger–Lambert–Gustavsson M2 brane proposal.

3. The exist various generalizations to higher Nambu brackets

$$ \{ f_1,\ldots ,f_n \} $$

with $n$ entries.

4. For more information, see this review.

1. There are already many good answers. Nevertheless, it seems appropriate in this connection to mention the Nambu bracket, which is a Poisson-like bracket

$$ \{ f,g,h \} $$

with 3 function entries, originally invented by Nambu in 1973, purportedly in a failed attempt to explain $SU(3)$ symmetry in quarks. Already Nambu discusses an operator version

$$ [\hat{f},\hat{g},\hat{h} ], $$

and one can imagine some kind of uncertainty relation associated to this, where canonical variables come in triplets. Unfortunately, the subject so far has remained just theoretical speculations. [Authors even don't agree what should replace the Jacobi identity for the Poisson bracket $\{ f,g\}$, although most think it should be the so-called Filippov fundamental Identity (FI).]

2. Recently in 2008, Nambu-brackets have been discussed in the Bagger–Lambert–Gustavsson M2 brane proposal.

3. The exist various generalizations to higher Nambu brackets

$$ \{ f_1,\ldots ,f_n \} $$

with $n$ entries.

4. For more information, see this review.