3 replaced http://physics.stackexchange.com/ with https://physics.stackexchange.com/
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Note first of all that complex conjugation does not just apply to bosonic variables. Grassmann numbers can also be complex conjugated, and one can speak of real and imaginary Grassmann numbers. So bosonic and fermionic variables behave similarly in that respect.

In fact, OP's question is not really about Grassmann numbers per se. The issue is rather whether a pair of variables is a complex conjugate pair or truly independent variables. This arises for bosonic coherent states as well, cf. e.g. thisthis Phys.SE post.

In Ref. 1 the variables $\eta$ and $\bar{\eta}$ are independent variables. Be aware that other authors may use different notations, such as e.g., $\eta$ and $\eta^{\ast}$.

References:

  1. T. Lancaster & S.J. Blundell, QFT for the Gifted Amateur, 2014; Section 28.2.

Note first of all that complex conjugation does not just apply to bosonic variables. Grassmann numbers can also be complex conjugated, and one can speak of real and imaginary Grassmann numbers. So bosonic and fermionic variables behave similarly in that respect.

In fact, OP's question is not really about Grassmann numbers per se. The issue is rather whether a pair of variables is a complex conjugate pair or truly independent variables. This arises for bosonic coherent states as well, cf. e.g. this Phys.SE post.

In Ref. 1 the variables $\eta$ and $\bar{\eta}$ are independent variables. Be aware that other authors may use different notations, such as e.g., $\eta$ and $\eta^{\ast}$.

References:

  1. T. Lancaster & S.J. Blundell, QFT for the Gifted Amateur, 2014; Section 28.2.

Note first of all that complex conjugation does not just apply to bosonic variables. Grassmann numbers can also be complex conjugated, and one can speak of real and imaginary Grassmann numbers. So bosonic and fermionic variables behave similarly in that respect.

In fact, OP's question is not really about Grassmann numbers per se. The issue is rather whether a pair of variables is a complex conjugate pair or truly independent variables. This arises for bosonic coherent states as well, cf. e.g. this Phys.SE post.

In Ref. 1 the variables $\eta$ and $\bar{\eta}$ are independent variables. Be aware that other authors may use different notations, such as e.g., $\eta$ and $\eta^{\ast}$.

References:

  1. T. Lancaster & S.J. Blundell, QFT for the Gifted Amateur, 2014; Section 28.2.
2 Minor edits
source | link

Note first of all that complex conjugation does not just apply to bosonic variables. Grassmann numbers can also be complex conjugated, and one can speak of real and imaginary Grassmann numbers. So bosonic and fermionic variables behave similarly in that respect.

In fact, OP's question is not really about Grassmann numbers per se. The issue is rather whether a pair of variables is a complex conjugate pair or truly independent variables. This arises for bosonic coherent states as well, cf. e.g. this Phys.SE post.

In Ref. 1 the variables $\eta$ and $\bar{\eta}$ are independent variables. Be aware that other authors mightmay use different notations, such as e.g., $\eta$ and $\eta^{\ast}$.

References:

T. Lancaster & S.J. Blundell, QFT for the Gifted Amateur, 2014; Section 28.2.

  1. T. Lancaster & S.J. Blundell, QFT for the Gifted Amateur, 2014; Section 28.2.

Note first of all that complex conjugation does not just apply to bosonic variables. Grassmann numbers can also be complex conjugated, and one can speak of real and imaginary Grassmann numbers. So bosonic and fermionic variables behave similarly in that respect.

In fact, OP's question is not really about Grassmann numbers per se. The issue is rather whether a pair of variables is a complex conjugate pair or truly independent variables. This arises for bosonic coherent states as well, cf. e.g. this Phys.SE post.

In Ref. 1 the variables $\eta$ and $\bar{\eta}$ are independent variables. Be aware that other authors might use different notations, such as e.g., $\eta$ and $\eta^{\ast}$.

References:

T. Lancaster & S.J. Blundell, QFT for the Gifted Amateur, 2014; Section 28.2.

Note first of all that complex conjugation does not just apply to bosonic variables. Grassmann numbers can also be complex conjugated, and one can speak of real and imaginary Grassmann numbers. So bosonic and fermionic variables behave similarly in that respect.

In fact, OP's question is not really about Grassmann numbers per se. The issue is rather whether a pair of variables is a complex conjugate pair or truly independent variables. This arises for bosonic coherent states as well, cf. e.g. this Phys.SE post.

In Ref. 1 the variables $\eta$ and $\bar{\eta}$ are independent variables. Be aware that other authors may use different notations, such as e.g., $\eta$ and $\eta^{\ast}$.

References:

  1. T. Lancaster & S.J. Blundell, QFT for the Gifted Amateur, 2014; Section 28.2.
1
source | link

Note first of all that complex conjugation does not just apply to bosonic variables. Grassmann numbers can also be complex conjugated, and one can speak of real and imaginary Grassmann numbers. So bosonic and fermionic variables behave similarly in that respect.

In fact, OP's question is not really about Grassmann numbers per se. The issue is rather whether a pair of variables is a complex conjugate pair or truly independent variables. This arises for bosonic coherent states as well, cf. e.g. this Phys.SE post.

In Ref. 1 the variables $\eta$ and $\bar{\eta}$ are independent variables. Be aware that other authors might use different notations, such as e.g., $\eta$ and $\eta^{\ast}$.

References:

T. Lancaster & S.J. Blundell, QFT for the Gifted Amateur, 2014; Section 28.2.