# Minkowski and Euclidean 4 vectors

I am reading these lecture notes. On page 114 they define Minkowskian space time coordinates:

$$\mathcal{X}=(\mathcal{X}^0,\mathcal{X}^1,\mathcal{X}^2,\mathcal{X}^3)=(t,x^i),$$

where $$x^1=x,x^2=y,x^3=z$$. They also define momenta:

$$\mathcal{K}=(\mathcal{K}^0,\mathcal{K}^1,\mathcal{K}^2,\mathcal{K}^3)=(k^0,k^i)$$

where $$k^1=k_x,k^2=k_y,k^3=k_z$$.

The Euclidean counterparts are defined as:

$$X=(X^0,X^1,X^2,X^3)=(\tau,x^i),$$

with $$\tau=i t$$

$$K=(K^0,K^1,K^2,K^3)=(k_n,k_i),$$

where $$k_n=-ik^0$$.

My first question is why don't we define $$K=(k_n,k^i)$$ ?

Next they calculatet in the notes $$X\cdot K$$. When I plug in all definitions I get:

$$X\cdot K=X_0K^0+X_iK^i=\tau k_n-X^iK^i=\tau k_n -x^i k_i=\tau k_n +x^i k^i=t k^0+\vec{k}\cdot\vec{x}$$

where $$\vec{x}=(x^1,x^2,x^3)$$ and $$\vec{k}=(k^1,k^2,k^3)$$ and I have used $$\eta_{\mu\nu}={\rm diag}(1,-1,-1,-1)$$.

Second question: In the lecture notes that I have linked they get a slightly different result:

$$X\cdot K=\tau k_n -\vec{k}\cdot\vec{x}=t k^0 -\vec{k}\cdot\vec{x}.$$

But this doesn't make sense because it is the same as $$\mathcal{X}\cdot\mathcal{K}$$ ???

You're making an error when using the Minkowski metric on the Euclidean components $$X$$ and $$K$$, which is why you get a different result. The result from the lecture notes is indeed correct and rightly in agreement with $$\mathcal{X}\cdot\mathcal{K}$$.
• Okay I see what you mean. $X\cdot K=X^0K^0+X^i K^i$ but $\mathcal{X}\cdot \mathcal{K}=\mathcal{X}^0\mathcal{K}^0-\mathcal{X}^i\mathcal{K}^i$. However I think there is still an inconsistency because $X\cdot X=X^0X^0+X^iX^i=\tau^2+x^ix^i=-t^2+\vec{x}\cdot \vec{x}$. Should this not be $X\cdot X=-t^2-\vec{x}\cdot\vec{x}$?