$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}$I have started to study particle physics, beginning with wikipedia and I am now reading David Griffiths "Introduction to Elementary Particles". I come from a background of applied mathematics and I have a beginners level understanding of representation theory, but I know the representation theory of $\SO_3$/$\O_3$ very well.

One question that haunted me in the beginning of my studies were the repeated mentions that some particle (like the Kaon or gluon) or some property (like the isospin) behaved according to some representation of a lie group, typically $\SU_2$ or $\SU_3$, but never was the vector space or even the action of the group on that vector space defined. How frustrating and intriguing at the same time!

For example, every quark has one of six colours (red, blue, green, antired, antiblue, antigreen) and a gluon can interact with a quark to give it a different colour, whereby a gluon carries one colour and one anti-colour. We would expect therefore, that there are 9 different types of gluons {(red,antired), (red,antigreen), ...}. However there only seem to exist 8, as the "Casimirelement" $$|9\rangle = (r,\bar r) + (b,\bar b) + (g,\bar g)$$ is not observed in our world (according to Griffiths, we would live in a very different world, if it did). Apparently, this is a representation of $\SU_3$.

Going back to a simpler example, let's look at spin and $\SU_2$: Many particles carry spin $\frac12$, and measuring the spin of it will either yield up or down spin ($\pm\frac12$), $$|\frac12\rangle := (1,0)^T, \hspace{2cm} |-\frac12\rangle := (0,1)^T.$$ However, prior to measuring the spin, the particle is in a superposition of these two states, so a general state is simply the vector $(\alpha,\beta)^T\in \mathbb{C}^2$, where we have the additional restraint that $$|\alpha|^2 + |\beta|^2 = 1.$$

We can now ask, how the vector $(\alpha,\beta)^T$ transforms, if we rotate the coordinate axis of our position space. For this we use the three Pauli spin matrices $\sigma_x, \sigma_y, \sigma_z\in\mathbb{C}^{2\times 2}$, put them in a 3-vector ${\sigma}:= (\sigma_x, \sigma_y, \sigma_z)^T$ and use the vector $\theta\in\mathbb{R}$ as rotational axis and its length for the angle of rotation. Then the transformation rule is given by

$$ (\alpha',\beta')^T = U(\theta) \cdot (\alpha, \beta)^T, \hspace{1cm} U(\theta):= e^{-\frac{i}{2} \theta\cdot\sigma}. $$ Here, I understand $\theta\cdot\sigma$ as the standard scalar product of $\mathbb{R}^3$, so we have a sum over the Pauli spin matrices, and hence have a $\mathbb{C}^{2\times 2}$ in the exponent.

This looks like it could be a representation of $\SO_3$, however I don't think so, for example rotating around the $z$-axis by $2\pi$ does not yield the same result as not rotating at all or rotating by $4\pi$.

Overall, though, the matrix $U(\theta)$ is an element of $\SU_2$, and hence we found a representation of $\SU_2$, which is closely related to $\SO_3$.

Finally, I come to my question:

• Is there more mathematical structure behind this transformation? This looks a little bit like a Lie-Algebra, but the exponential map does not quite fit ($\forall X \in \text{ Lie-Algebra : } \forall t\in \mathbb{R}: e^{tX}\in \text{ Lie-Group}$).

$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}$I have started to study particle physics, beginning with wikipedia and I am now reading David Griffiths "Introduction to Elementary Particles". I come from a background of applied mathematics and I have a beginners level understanding of representation theory, but I know the representation theory of $\SO_3$/$\O_3$ very well.

One question that haunted me in the beginning of my studies were the repeated mentions that some particle (like the Kaon or gluon) or some property (like the isospin) behaved according to some representation of a lie group, typically $\SU_2$ or $\SU_3$, but never was the vector space or even the action of the group on that vector space defined. How frustrating and intriguing at the same time!

For example, every quark has one of six colours (red, blue, green, antired, antiblue, antigreen) and a gluon can interact with a quark to give it a different colour, whereby a gluon carries one colour and one anti-colour. We would expect therefore, that there are 9 different types of gluons {(red,antired), (red,antigreen), ...}. However there only seem to exist 8, as the "Casimirelement" $$|9\rangle = (r,\bar r) + (b,\bar b) + (g,\bar g)$$ is not observed in our world (according to Griffiths, we would live in a very different world, if it did). Apparently, this is a representation of $\SU_3$.

Going back to a simpler example, let's look at spin and $\SU_2$: Many particles carry spin $\frac12$, and measuring the spin of it will either yield up or down spin ($\pm\frac12$), $$|\frac12\rangle := (1,0)^T, \hspace{2cm} |-\frac12\rangle := (0,1)^T.$$ However, prior to measuring the spin, the particle is in a superposition of these two states, so a general state is simply the vector $(\alpha,\beta)^T\in \mathbb{C}^2$, where we have the additional restraint that $$|\alpha|^2 + |\beta|^2 = 1.$$

We can now ask, how the vector $(\alpha,\beta)^T$ transforms, if we rotate the coordinate axis of our position space. For this we use the three Pauli spin matrices $\sigma_x, \sigma_y, \sigma_z\in\mathbb{C}^{2\times 2}$, put them in a 3-vector ${\sigma}:= (\sigma_x, \sigma_y, \sigma_z)^T$ and use the vector $\theta\in\mathbb{R}$ as rotational axis and its length for the angle of rotation. Then the transformation rule is given by

$$ (\alpha',\beta')^T = U(\theta) \cdot (\alpha, \beta)^T, \hspace{1cm} U(\theta):= e^{-\frac{i}{2} \theta\cdot\sigma}. $$ Here, I understand $\theta\cdot\sigma$ as the standard scalar product of $\mathbb{R}^3$, so we have a sum over the Pauli spin matrices, and hence have a $\mathbb{C}^{2\times 2}$-matrix in the exponent.

This looks like it could be a representation of $\SO_3$, however I don't think so, for example rotating around the $z$-axis by $2\pi$ does not yield the same result as not rotating at all or rotating by $4\pi$.

Overall, though, the matrix $U(\theta)$ is an element of $\SU_2$, and hence we found a representation of $\SU_2$, which is closely related to $\SO_3$.

Finally, I come to my question:

• Is there more mathematical structure behind this transformation? This looks a little bit like a Lie-Algebra, but the exponential map does not quite fit ($\forall X \in \text{ Lie-Algebra : } \forall t\in \mathbb{R}: e^{tX}\in \text{ Lie-Group}$).

$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}$I have started to study particle physics, beginning with wikipedia and I am now reading David Griffiths "Introduction to Elementary Particles". I come from a background of applied mathematics and I have a beginners level understanding of representation theory, but I know the representation theory of $\SO_3$/$\O_3$ very well.

One question that haunted me in the beginning of my studies were the repeated mentions that some particle (like the Kaon or gluon) or some property (like the isospin) behaved according to some representation of a lie group, typically $\SU_2$ or $\SU_3$, but never was the vector space or even the action of the group on that vector space defined. How frustrating and intriguing at the same time!

For example, every quark has one of six colours (red, blue, green, antired, antiblue, antigreen) and a gluon can interact with a quark to give it a different colour, whereby a gluon carries one colour and one anti-colour. We would expect therefore, that there are 9 different types of gluons {(red,antired), (red,antigreen), ...}. However there only seem to exist 8, as the "Casimirelement" $$|9\rangle = (r,\bar r) + (b,\bar b) + (g,\bar g)$$ is not observed in our world (according to Griffiths, we would live in a very different world, if it did). Apparently, this is a representation of $\SU_3$.

Going back to a simpler example, let's look at spin and $\SU_2$: Many particles carry spin $\frac12$, and measuring the spin of it will either yield up or down spin ($\pm\frac12$), $$|\frac12\rangle := (1,0)^T, \hspace{2cm} |-\frac12\rangle := (0,1)^T.$$ However, prior to measuring the spin, the particle is in a superposition of these two states, so a general state is simply the vector $(\alpha,\beta)^T\in \mathbb{C}^2$, where we have the additional restraint that $$|\alpha|^2 + |\beta|^2 = 1.$$

We can now ask, how the vector $(\alpha,\beta)^T$ transforms, if we rotate the coordinate axis of our position space. For this we use the three Pauli spin matrices $\sigma_x, \sigma_y, \sigma_z\in\mathbb{C}^{2\times 2}$, put them in a 3-vector ${\sigma}:= (\sigma_x, \sigma_y, \sigma_z)^T$ and use the vector $\theta\in\mathbb{R}$ as rotational axis and its length for the angle of rotation. Then the transformation rule is given by

$$ (\alpha',\beta')^T = U(\theta) \cdot (\alpha, \beta)^T, \hspace{1cm} U(\theta):= e^{-\frac{i}{2} \theta\cdot\sigma}. $$ Here, I understand $\theta\cdot\sigma$ as the standard scalar product of $\mathbb{R}^3$, so we have a sum over the Pauli spin matrices, and hence have a $\mathbb{C}^{2\times 2}$ in the exponent.

This looks like it could be a representation of $\SO_3$, however I don't think so, for example rotating around the $z$-axis by $2\pi$ does not yield the same result as not rotating at all or rotating by $4\pi$.

Overall, though, we the matrix $U(\theta)$ is an element of $\SU_2$, and hence we found a representation of $\SU_2$, which is closely related to $\SO_3$.

Finally, I come to my question:

• Is there more mathematical structure behind this transformation? This looks a little bit like a Lie-Algebra, but the exponential map does not quite fit ($\forall X \in \text{ Lie-Algebra : } \forall t\in \mathbb{R}: e^{tX}\in \text{ Lie-Group}$).

$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}$I have started to study particle physics, beginning with wikipedia and I am now reading David Griffiths "Introduction to Elementary Particles". I come from a background of applied mathematics and I have a beginners level understanding of representation theory, but I know the representation theory of $\SO_3$/$\O_3$ very well.

One question that haunted me in the beginning of my studies were the repeated mentions that some particle (like the Kaon or gluon) or some property (like the isospin) behaved according to some representation of a lie group, typically $\SU_2$ or $\SU_3$, but never was the vector space or even the action of the group on that vector space defined. How frustrating and intriguing at the same time!

For example, every quark has one of six colours (red, blue, green, antired, antiblue, antigreen) and a gluon can interact with a quark to give it a different colour, whereby a gluon carries one colour and one anti-colour. We would expect therefore, that there are 9 different types of gluons {(red,antired), (red,antigreen), ...}. However there only seem to exist 8, as the "Casimirelement" $$|9\rangle = (r,\bar r) + (b,\bar b) + (g,\bar g)$$ is not observed in our world (according to Griffiths, we would live in a very different world, if it did). Apparently, this is a representation of $\SU_3$.

Going back to a simpler example, let's look at spin and $\SU_2$: Many particles carry spin $\frac12$, and measuring the spin of it will either yield up or down spin ($\pm\frac12$), $$|\frac12\rangle := (1,0)^T, \hspace{2cm} |-\frac12\rangle := (0,1)^T.$$ However, prior to measuring the spin, the particle is in a superposition of these two states, so a general state is simply the vector $(\alpha,\beta)^T\in \mathbb{C}^2$, where we have the additional restraint that $$|\alpha|^2 + |\beta|^2 = 1.$$

We can now ask, how the vector $(\alpha,\beta)^T$ transforms, if we rotate the coordinate axis of our position space. For this we use the three Pauli spin matrices $\sigma_x, \sigma_y, \sigma_z\in\mathbb{C}^{2\times 2}$, put them in a 3-vector ${\sigma}:= (\sigma_x, \sigma_y, \sigma_z)^T$ and use the vector $\theta\in\mathbb{R}$ as rotational axis and its length for the angle of rotation. Then the transformation rule is given by

$$ (\alpha',\beta')^T = U(\theta) \cdot (\alpha, \beta)^T, \hspace{1cm} U(\theta):= e^{-\frac{i}{2} \theta\cdot\sigma}. $$ Here, I understand $\theta\cdot\sigma$ as the standard scalar product of $\mathbb{R}^3$, so we have a sum over the Pauli spin matrices, and hence have a $\mathbb{C}^{2\times 2}$ in the exponent.

This looks like it could be a representation of $\SO_3$, however I don't think so, for example rotating around the $z$-axis by $2\pi$ does not yield the same result as not rotating at all or rotating by $4\pi$.

Overall, though, the matrix $U(\theta)$ is an element of $\SU_2$, and hence we found a representation of $\SU_2$, which is closely related to $\SO_3$.

Finally, I come to my question:

• Is there more mathematical structure behind this transformation? This looks a little bit like a Lie-Algebra, but the exponential map does not quite fit ($\forall X \in \text{ Lie-Algebra : } \forall t\in \mathbb{R}: e^{tX}\in \text{ Lie-Group}$).

I have started to study particle physics, beginning with wikipedia and I am now reading David Griffiths "Introduction to Elementary Particles". I come from a background of applied mathematics and I have a beginners level understanding of representation theory, but I know the representation theory of $SO_3$/$O_3$ very well.

One question that haunted me in the beginning of my studies were the repeated mentions that some particle (like the Kaon or gluon) or some property (like the isospin) behaved according to some representation of a lie group, typically $SU_2$ or $SU_3$, but never was the vector space or even the action of the group on that vector space defined. How frustrating and intriguing at the same time!

For example, every quark has one of six colours (red, blue, green, antired, antiblue, antigreen) and a gluon can interact with a quark to give it a different colour, whereby a gluon carries one colour and one anti-colour. We would expect therefore, that there are 9 different types of gluons {(red,antired), (red,antigreen), ...}. However there only seem to exist 8, as the "Casimirelement" $$|9> = (r,\bar r) + (b,\bar b) + (g,\bar g)$$ is not observed in our world (according to Griffiths, we would live in a very different world, if it did). Apparently, this is a representation of $SU_3$.

Going back to a simpler example, let's look at spin and $SU_2$: Many particles carry spin $\frac12$, and measuring the spin of it will either yield up or down spin ($\pm\frac12$), $$|\frac12> := (1,0)^T, \hspace{2cm} |-\frac12> := (0,1)^T.$$ However, prior to measuring the spin, the particle is in a superposition of these two states, so a general state is simply the vector $(\alpha,\beta)^T\in \mathbb{C}^2$, where we have the additional restraint that $$|\alpha|^2 + |\beta|^2 = 1.$$

We can now ask, how the vector $(\alpha,\beta)^T$ transforms, if we rotate the coordinate axis of our position space. For this we use the three Pauli spin matrices $\sigma_x, \sigma_y, \sigma_z\in\mathbb{C}^{2\times 2}$, put them in a 3-vector ${\sigma}:= (\sigma_x, \sigma_y, \sigma_z)^T$ and use the vector $\theta\in\mathbb{R}$ as rotational axis and its length for the angle of rotation. Then the transformation rule is given by

$$ (\alpha',\beta')^T = U(\theta) \cdot (\alpha, \beta)^T, \hspace{1cm} U(\theta):= e^{-\frac{i}{2} \theta\cdot\sigma}. $$ Here, I understand $\theta\cdot\sigma$ as the standard scalar product of $\mathbb{R}^3$, so we have a sum over the Pauli spin matrices, and hence have a $\mathbb{C}^{2\times 2}$ in the exponent.

This looks like it could be a representation of $SO_3$, however I don't think so, for example rotating around the $z$-axis by $2\pi$ does not yield the same result as not rotating at all or rotating by $4\pi$.

Overall, though, we the matrix $U(\theta)$ is an element of $SU_2$, and hence we found a representation of $SU_2$, which is closely related to $SO_3$.

Finally, I come to my question:

• Is there more mathematical structure behind this transformation? This looks a little bit like a Lie-Algebra, but the exponential map does not quite fit ($\forall X \in \text{ Lie-Algebra : } \forall t\in \mathbb{R}: e^{tX}\in \text{ Lie-Group}$).

$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}$I have started to study particle physics, beginning with wikipedia and I am now reading David Griffiths "Introduction to Elementary Particles". I come from a background of applied mathematics and I have a beginners level understanding of representation theory, but I know the representation theory of $\SO_3$/$\O_3$ very well.

One question that haunted me in the beginning of my studies were the repeated mentions that some particle (like the Kaon or gluon) or some property (like the isospin) behaved according to some representation of a lie group, typically $\SU_2$ or $\SU_3$, but never was the vector space or even the action of the group on that vector space defined. How frustrating and intriguing at the same time!

For example, every quark has one of six colours (red, blue, green, antired, antiblue, antigreen) and a gluon can interact with a quark to give it a different colour, whereby a gluon carries one colour and one anti-colour. We would expect therefore, that there are 9 different types of gluons {(red,antired), (red,antigreen), ...}. However there only seem to exist 8, as the "Casimirelement" $$|9\rangle = (r,\bar r) + (b,\bar b) + (g,\bar g)$$ is not observed in our world (according to Griffiths, we would live in a very different world, if it did). Apparently, this is a representation of $\SU_3$.

Going back to a simpler example, let's look at spin and $\SU_2$: Many particles carry spin $\frac12$, and measuring the spin of it will either yield up or down spin ($\pm\frac12$), $$|\frac12\rangle := (1,0)^T, \hspace{2cm} |-\frac12\rangle := (0,1)^T.$$ However, prior to measuring the spin, the particle is in a superposition of these two states, so a general state is simply the vector $(\alpha,\beta)^T\in \mathbb{C}^2$, where we have the additional restraint that $$|\alpha|^2 + |\beta|^2 = 1.$$

We can now ask, how the vector $(\alpha,\beta)^T$ transforms, if we rotate the coordinate axis of our position space. For this we use the three Pauli spin matrices $\sigma_x, \sigma_y, \sigma_z\in\mathbb{C}^{2\times 2}$, put them in a 3-vector ${\sigma}:= (\sigma_x, \sigma_y, \sigma_z)^T$ and use the vector $\theta\in\mathbb{R}$ as rotational axis and its length for the angle of rotation. Then the transformation rule is given by

$$ (\alpha',\beta')^T = U(\theta) \cdot (\alpha, \beta)^T, \hspace{1cm} U(\theta):= e^{-\frac{i}{2} \theta\cdot\sigma}. $$ Here, I understand $\theta\cdot\sigma$ as the standard scalar product of $\mathbb{R}^3$, so we have a sum over the Pauli spin matrices, and hence have a $\mathbb{C}^{2\times 2}$ in the exponent.

This looks like it could be a representation of $\SO_3$, however I don't think so, for example rotating around the $z$-axis by $2\pi$ does not yield the same result as not rotating at all or rotating by $4\pi$.

Overall, though, we the matrix $U(\theta)$ is an element of $\SU_2$, and hence we found a representation of $\SU_2$, which is closely related to $\SO_3$.

Finally, I come to my question:

• Is there more mathematical structure behind this transformation? This looks a little bit like a Lie-Algebra, but the exponential map does not quite fit ($\forall X \in \text{ Lie-Algebra : } \forall t\in \mathbb{R}: e^{tX}\in \text{ Lie-Group}$).